Turn continuous equations into finite-dimensional approximations.
Workflow stages are a navigation guide; numerical techniques often span several stages. Select a technique to open its description; use “Go back to the slider” to return here.
HOW THE TECHNIQUES CONNECT
A tree of numerical techniques.
Explore workflow stage → method family → technique. Branches organize all 68 entries; select a technique to see its description and relationships. Connections between families appear in each technique’s “Relationships to other techniques” section.
This is a browsing hierarchy, not a universal dependency tree. “Same discipline” means shared subject area; specific links distinguish approximations, extensions, closures, numerical methods, and coupling.
A curated, expandable atlas of numerical techniques. It is not exhaustive. Workflow labels indicate a typical role, and techniques can be combined across stages.
Educational reference only. Verify assumptions and results before practical use. Read the legal disclaimer.
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Approximates derivatives with weighted values on a grid.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
u′′(xi)≈h2ui+1−2ui+ui−1
Derivation / construction sketch
Expand neighboring values in Taylor series.
Add the expansions to cancel odd derivatives.
Divide by h squared to obtain the centered second derivative.
Symbols & assumptions
Second-order accuracy requires a smooth solution and a uniform grid; boundaries need compatible stencils.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Implementation route: assemble stencils in Python/NumPy or a compiled solver; this reference is a textbook, not a packaged application.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Temperature fields in heat sinks and plates.
Method limitations
Second-order accuracy requires a smooth solution and a uniform grid; boundaries need compatible stencils.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balances conserved quantities over control volumes.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
VidtdUi+f∑Ff⋅nfAf=ViSi
Derivation / construction sketch
Integrate the conservation law over a cell.
Use the divergence theorem to convert volume flux divergence to surface flux.
Approximate each face flux consistently with its neighbor.
Symbols & assumptions
Accuracy depends on reconstruction, numerical flux, mesh quality, and time integration.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Pipe-flow and aerodynamic CFD workflows.
Method limitations
Accuracy depends on reconstruction, numerical flux, mesh quality, and time integration.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses piecewise basis functions and a weak formulation on a mesh.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
uh=j∑UjNjKij=∫Ω∇Ni⋅k∇NjdΩKU=f
Derivation / construction sketch
Multiply a diffusion equation by a test function.
Integrate by parts and impose boundary conditions.
Expand in basis functions and assemble the element contributions.
Symbols & assumptions
Diffusion example shown; elements and function spaces must suit the PDE and boundary conditions.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Spatial discretization
Practical use
Structural brackets, heat exchangers, and electromagnetic components.
Engineering application examples
Structural brackets, heat exchangers, and electromagnetic components.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Structural brackets, heat exchangers, and electromagnetic components.
Method limitations
Diffusion example shown; elements and function spaces must suit the PDE and boundary conditions.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines element-local trial functions with numerical fluxes across interfaces.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
∫Kv∂tuh−∫K∇v⋅F(uh)+∫∂KvF⋅n=0
Derivation / construction sketch
Use independent polynomial spaces on each element.
Integrate the conservation law against local test functions.
Couple neighbors through a stable numerical flux.
Symbols & assumptions
Flux choice and stabilization matter; explicit high-order schemes can need small time steps.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Spatial discretization
Practical use
Compressible-flow simulation for nozzles and wings.
Engineering application examples
Compressible-flow simulation for nozzles and wings.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Compressible-flow simulation for nozzles and wings.
Method limitations
Flux choice and stabilization matter; explicit high-order schemes can need small time steps.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Transfers suitable linear PDE problems to boundary integral equations.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
c(x)u(x)+∫Γu∂nGdΓ=∫ΓG∂nudΓ
Derivation / construction sketch
Apply Green's identity with a fundamental solution.
Take the observation point to the boundary.
Discretize the boundary fields and integrals.
Symbols & assumptions
Representative Laplace formulation; singular quadrature and dense matrices require care.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Spatial discretization
Practical use
Exterior acoustics, electrostatics, and scattering models.
Engineering application examples
Exterior acoustics, electrostatics, and scattering models.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Exterior acoustics, electrostatics, and scattering models.
Method limitations
Representative Laplace formulation; singular quadrature and dense matrices require care.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates smooth fields globally and enforces the equation at selected nodes.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
uN(x)=j=0∑NUjℓj(x)uN′(xi)=j∑DijUj
Derivation / construction sketch
Interpolate a field with a global polynomial or Fourier basis.
Differentiate the basis analytically.
Enforce the PDE at collocation nodes.
Symbols & assumptions
Rapid convergence requires sufficient smoothness; discontinuities cause oscillations and aliasing.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Dedalus provides spectral methods; the paper discusses its formulation and implementation. Collocation details depend on basis and problem.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Smooth fluid instabilities and wave propagation.
Method limitations
Rapid convergence requires sufficient smoothness; discontinuities cause oscillations and aliasing.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Builds meshfree interpolants or derivative stencils from radial kernels.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
uh(x)=j∑cjϕ(∥x−xj∥)(LΦ)c=f
Derivation / construction sketch
Choose kernel centers and a radial basis.
Apply the differential operator to the basis functions.
Collocate interior and boundary equations to solve for coefficients.
Symbols & assumptions
Polynomial augmentation and shape parameters affect solvability and conditioning.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Spatial discretization
Practical use
Scattered geometry interpolation and meshfree field approximation.
Engineering application examples
Scattered geometry interpolation and meshfree field approximation.
Interpolation implementation; a PDE collocation solver requires applying operators and boundary conditions separately.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Scattered geometry interpolation and meshfree field approximation.
Method limitations
Polynomial augmentation and shape parameters affect solvability and conditioning.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Concentrates degrees of freedom where a numerical error indicator is large.
DiscretizeNumerical technique
Formulation & short derivation
Representative numerical formulation
η2=K∑ηK2K∈M∑ηK2≥θη2
Derivation / construction sketch
Solve on the current mesh.
Estimate local error and mark cells carrying a chosen error fraction.
Refine, transfer the solution, and repeat.
Symbols & assumptions
Bulk-marking criterion shown; an indicator is not automatically a rigorous error bound.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Spatial discretization
Practical use
Resolving stress concentrations and localized heat sources.
Engineering application examples
Resolving stress concentrations and localized heat sources.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Resolving stress concentrations and localized heat sources.
Method limitations
Bulk-marking criterion shown; an indicator is not automatically a rigorous error bound.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solves a linear system through triangular factors with pivoting.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
PA=LULy=PbUx=y
Derivation / construction sketch
Eliminate entries below the diagonal.
Record elimination multipliers in a lower triangular factor.
Apply the row permutation and solve two triangular systems.
Symbols & assumptions
Pivoting is important; sparse fill-in and ill-conditioning can dominate cost and accuracy.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Circuit matrices and repeated small-to-medium engineering solves.
Engineering application examples
Circuit matrices and repeated small-to-medium engineering solves.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Circuit matrices and repeated small-to-medium engineering solves.
Method limitations
Pivoting is important; sparse fill-in and ill-conditioning can dominate cost and accuracy.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Factors a symmetric positive-definite matrix efficiently.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
A=LLTLy=bLTx=y
Derivation / construction sketch
Match entries in the product of a lower triangular factor and its transpose.
Compute each positive diagonal square root.
Use forward and backward substitution.
Symbols & assumptions
Requires positive definiteness; the complex analogue uses a conjugate transpose.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Elasticity systems with sufficient constraints and covariance calculations.
Engineering application examples
Elasticity systems with sufficient constraints and covariance calculations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Elasticity systems with sufficient constraints and covariance calculations.
Method limitations
Requires positive definiteness; the complex analogue uses a conjugate transpose.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses an orthogonal factorization to solve least-squares systems.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
A=QRRx=QTb
Derivation / construction sketch
Apply orthogonal transformations to eliminate subdiagonal entries.
Orthogonality preserves the residual norm.
Solve the triangular least-squares problem.
Symbols & assumptions
Thin full-rank form shown; rank-deficient problems need pivoting or SVD.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Sensor calibration and polynomial parameter fitting.
Engineering application examples
Sensor calibration and polynomial parameter fitting.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Sensor calibration and polynomial parameter fitting.
Method limitations
Thin full-rank form shown; rank-deficient problems need pivoting or SVD.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Separates matrix directions by their amplification strengths.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
A=UΣVTx=VΣ+UTb
Derivation / construction sketch
Diagonalize the action of the matrix into orthogonal input and output directions.
Invert retained nonzero singular values.
Discard or regularize poorly determined directions if required.
Symbols & assumptions
Threshold choice affects effective rank; SVD can be expensive for large dense systems.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Inverse imaging, reduced-order models, and signal denoising.
Engineering application examples
Inverse imaging, reduced-order models, and signal denoising.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Inverse imaging, reduced-order models, and signal denoising.
Method limitations
Threshold choice affects effective rank; SVD can be expensive for large dense systems.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Builds crystal electronic bands from localized orbitals and intersite hopping.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Predicts band gaps by perturbing free electrons with a weak periodic potential.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Computes phonon modes from a quadratic expansion of crystal potential energy.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Updates each unknown from the previous iterate using the diagonal.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
A=D+L+Uxk+1=D−1[b−(L+U)xk]
Derivation / construction sketch
Split the matrix into diagonal and off-diagonal parts.
Move the off-diagonal contribution to the right-hand side.
Evaluate all new components from the old iterate.
Symbols & assumptions
Convergence requires the iteration matrix spectral radius below one; diagonal dominance is a sufficient condition.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Parallel smoothing and simple stationary iterative solvers.
Engineering application examples
Parallel smoothing and simple stationary iterative solvers.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Parallel smoothing and simple stationary iterative solvers.
Method limitations
Convergence requires the iteration matrix spectral radius below one; diagonal dominance is a sufficient condition.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses newly updated values immediately within each sweep.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
(D+L)xk+1=b−Uxk
Derivation / construction sketch
Split the matrix into lower triangular and upper parts.
Sweep through the unknowns in a fixed order.
Reuse new components as soon as they are available.
Symbols & assumptions
Ordering influences convergence and parallelism; convergence is not guaranteed for arbitrary matrices.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Smoothers inside multigrid and structured-grid elliptic solvers.
Engineering application examples
Smoothers inside multigrid and structured-grid elliptic solvers.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Smoothers inside multigrid and structured-grid elliptic solvers.
Method limitations
Ordering influences convergence and parallelism; convergence is not guaranteed for arbitrary matrices.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relaxes a Gauss-Seidel correction with a tunable weight.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
xk+1=(D+ωL)−1[ωb−((ω−1)D+ωU)xk]
Derivation / construction sketch
Form the Gauss-Seidel component update.
Blend its correction using relaxation weight omega.
Choose the weight to improve convergence for the problem class.
Symbols & assumptions
For symmetric positive-definite systems, 0 < omega < 2 gives convergence; an optimal weight is problem dependent.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Elliptic potential and diffusion calculations.
Method limitations
For symmetric positive-definite systems, 0 < omega < 2 gives convergence; an optimal weight is problem dependent.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Minimize the quadratic energy along a search direction.
Choose successive directions to be A-conjugate.
Update residuals and stop using a scaled tolerance.
Symbols & assumptions
Unpreconditioned step shown; A must be symmetric positive definite, with compatible preconditioning.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Large constrained elasticity and diffusion systems.
Engineering application examples
Large constrained elasticity and diffusion systems.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large constrained elasticity and diffusion systems.
Method limitations
Unpreconditioned step shown; A must be symmetric positive definite, with compatible preconditioning.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Minimizes the residual over a Krylov subspace for nonsymmetric systems.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
xm=x0+Vmymym=argymin∥βe1−Hˉmy∥2
Derivation / construction sketch
Build an orthonormal Krylov basis with Arnoldi iteration.
Represent the matrix action by an upper Hessenberg matrix.
Solve the small residual-minimization problem.
Symbols & assumptions
Restarting limits storage but may slow or stall convergence; preconditioning is often essential.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Advection-diffusion and coupled multiphysics linearizations.
Engineering application examples
Advection-diffusion and coupled multiphysics linearizations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Advection-diffusion and coupled multiphysics linearizations.
Method limitations
Restarting limits storage but may slow or stall convergence; preconditioning is often essential.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses short recurrences to stabilize a nonsymmetric Krylov iteration.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
rk=pk(A)r0pk(0)=1
Derivation / construction sketch
Construct a bi-conjugate residual polynomial.
Combine it with local residual-smoothing factors.
Use short vector recurrences to avoid storing a full Krylov basis.
Symbols & assumptions
Polynomial viewpoint only; breakdown and irregular convergence are possible.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Memory-limited nonsymmetric CFD matrix solves.
Method limitations
Polynomial viewpoint only; breakdown and irregular convergence are possible.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Restrict the residual and solve for coarse-grid error.
Prolong the correction and apply additional smoothing.
Symbols & assumptions
Transfers, coarse operators, boundary conditions, and smoothers must work together.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Large Poisson and diffusion problems on mesh hierarchies.
Engineering application examples
Large Poisson and diffusion problems on mesh hierarchies.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large Poisson and diffusion problems on mesh hierarchies.
Method limitations
Transfers, coarse operators, boundary conditions, and smoothers must work together.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Constructs coarse spaces from matrix structure rather than an explicit mesh hierarchy.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
Ac=RAPx←x+PAc−1R(b−Ax)
Derivation / construction sketch
Identify algebraically strong connections.
Construct interpolation and a coarse operator.
Combine coarse corrections with relaxation.
Symbols & assumptions
One correction shown; performance depends on operator structure and coarsening choices.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Linear algebra
Practical use
Large sparse elliptic systems on complex engineering meshes.
Engineering application examples
Large sparse elliptic systems on complex engineering meshes.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large sparse elliptic systems on complex engineering meshes.
Method limitations
One correction shown; performance depends on operator structure and coarsening choices.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Reliably narrows a continuous scalar root bracket.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
ck=2ak+bkbk−ak=2kb0−a0
Derivation / construction sketch
Begin with opposite endpoint signs.
Evaluate the midpoint.
Retain the half interval that preserves a sign change.
Symbols & assumptions
Continuity and a valid bracket are required; even-multiplicity roots may not change sign.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Finding operating points from monotone balance equations.
Engineering application examples
Finding operating points from monotone balance equations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Finding operating points from monotone balance equations.
Method limitations
Continuity and a valid bracket are required; even-multiplicity roots may not change sign.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solves nonlinear equations by repeated local linearization.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
J(xk)sk=−F(xk)xk+1=xk+sk
Derivation / construction sketch
Expand the residual to first order at the current state.
Set the linearized residual to zero.
Solve for a correction and repeat, with damping if needed.
Symbols & assumptions
Fast local convergence requires a suitable initial guess, smoothness, and a nonsingular Jacobian.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Nonlinear material equilibria and implicit time-step equations.
Engineering application examples
Nonlinear material equilibria and implicit time-step equations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Nonlinear material equilibria and implicit time-step equations.
Method limitations
Fast local convergence requires a suitable initial guess, smoothness, and a nonsingular Jacobian.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates a scalar derivative from two previous points.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
xk+1=xk−f(xk)f(xk)−f(xk−1)xk−xk−1
Derivation / construction sketch
Replace Newton's derivative with a divided difference.
Intersect the resulting secant line with the horizontal axis.
Advance the pair of iterates.
Symbols & assumptions
Does not preserve a root bracket; small denominator differences can cause large steps.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Scalar balance equations with expensive derivatives.
Engineering application examples
Scalar balance equations with expensive derivatives.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Scalar balance equations with expensive derivatives.
Method limitations
Does not preserve a root bracket; small denominator differences can cause large steps.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines bracket reliability with interpolation-based acceleration.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
f(ak)f(bk)≤0ck∈(ak,bk)
Derivation / construction sketch
Maintain a sign-changing bracket.
Try secant or inverse-quadratic interpolation when its step is acceptable.
Fall back to bisection when interpolation is unsafe.
Symbols & assumptions
Invariant shown rather than a full algorithm; the function must be continuous on a valid bracket.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Robust engineering threshold and equilibrium calculations.
Engineering application examples
Robust engineering threshold and equilibrium calculations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Robust engineering threshold and equilibrium calculations.
Method limitations
Invariant shown rather than a full algorithm; the function must be continuous on a valid bracket.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Approximates the compressibility factor of a monodisperse hard-sphere fluid from its packing fraction.
Invert the hard-sphere EOS for packing fraction at a specified pressure using a bracket within the fluid regime. Direct pressure evaluation requires no root solver.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Iterates a rearranged equation until the state stops changing.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
xk+1=g(xk)∥g(x)−g(y)∥≤q∥x−y∥,q<1
Derivation / construction sketch
Rewrite the equation as x = g(x).
Use the latest state on the right-hand side.
A contraction on an invariant complete set guarantees convergence.
Symbols & assumptions
A poor rearrangement can diverge even when the original equation has a root.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Partitioned coupling and simple nonlinear balance solvers.
Engineering application examples
Partitioned coupling and simple nonlinear balance solvers.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Partitioned coupling and simple nonlinear balance solvers.
Method limitations
A poor rearrangement can diverge even when the original equation has a root.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Take a step using the current Jacobian approximation.
Measure the residual change.
Apply a rank-one correction satisfying the new secant condition.
Symbols & assumptions
Good-Broyden Jacobian form shown; scaling, safeguards, and initial approximation matter.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Nonlinear systems when full Jacobians are costly.
Method limitations
Good-Broyden Jacobian form shown; scaling, safeguards, and initial approximation matter.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solves each Newton correction approximately with a Krylov method.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
J(x)v≈εF(x+εv)−F(x)Js=−F(x)
Derivation / construction sketch
Linearize the nonlinear residual.
Supply Jacobian-vector products without necessarily forming a matrix.
Use a preconditioned Krylov solve and globalize the Newton step.
Symbols & assumptions
Finite-difference perturbations balance truncation and roundoff; preconditioning remains critical.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Large nonlinear PDE systems and implicit multiphysics.
Engineering application examples
Large nonlinear PDE systems and implicit multiphysics.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large nonlinear PDE systems and implicit multiphysics.
Method limitations
Finite-difference perturbations balance truncation and roundoff; preconditioning remains critical.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tracks solution branches through turning points by augmenting the nonlinear system.
Solve algebraNumerical technique
Formulation & short derivation
Representative numerical formulation
F(x,λ)=0txT(x−x0)+tλ(λ−λ0)=Δs
Derivation / construction sketch
Compute a tangent to the known solution branch.
Predict along that tangent.
Correct using the original residual and an arclength constraint.
Symbols & assumptions
Step-size adaptation and branch switching need additional logic; the constraint is local.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Nonlinear equations
Practical use
Buckling, bifurcation, and nonlinear operating-envelope analysis.
Engineering application examples
Buckling, bifurcation, and nonlinear operating-envelope analysis.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Buckling, bifurcation, and nonlinear operating-envelope analysis.
Method limitations
Step-size adaptation and branch switching need additional logic; the constraint is local.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximate the integral with its left-endpoint slope.
Repeat with a step size that satisfies stability and accuracy needs.
Symbols & assumptions
First-order global accuracy; the explicit stability region is limited.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Simple dynamics prototypes and teaching simulations.
Engineering application examples
Simple dynamics prototypes and teaching simulations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Simple dynamics prototypes and teaching simulations.
Method limitations
First-order global accuracy; the explicit stability region is limited.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Balances thermal storage, conduction and heat sources.
For a nonstiff ODE or semidiscrete equation when the explicit stability bound and error budget permit; usually a baseline rather than the most efficient choice.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses the next-step slope and solves an implicit equation.
Advance in timeNumerical technique
Formulation & short derivation
Representative numerical formulation
yn+1=yn+hf(tn+1,yn+1)
Derivation / construction sketch
Approximate the time integral with its right-endpoint slope.
Rearrange as a nonlinear residual for the new state.
Solve that residual each step.
Symbols & assumptions
First order and strongly damping; an implicit solve does not guarantee an accurate large step.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Stiff thermal and dissipative systems.
Method limitations
First order and strongly damping; an implicit solve does not guarantee an accurate large step.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Averages endpoint slopes to obtain a second-order implicit step.
Advance in timeNumerical technique
Formulation & short derivation
Representative numerical formulation
yn+1=yn+2h[f(tn,yn)+f(tn+1,yn+1)]
Derivation / construction sketch
Integrate over one time step.
Use trapezoidal quadrature for the right-hand side.
Solve the resulting implicit system.
Symbols & assumptions
Second order for smooth solutions; A-stable for the test equation but not L-stable, so stiff oscillations can persist.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Transient diffusion and parabolic PDE discretizations.
Engineering application examples
Transient diffusion and parabolic PDE discretizations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Transient diffusion and parabolic PDE discretizations.
Method limitations
Second order for smooth solutions; A-stable for the test equation but not L-stable, so stiff oscillations can persist.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Evaluate beginning, midpoint, and endpoint slopes.
Choose weights to match the Taylor expansion through fourth order.
Combine the stages to update the state.
Symbols & assumptions
Explicit and not suited to strongly stiff problems without very small steps; no embedded error estimate.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Nonstiff flight, vibration, and control simulations.
Engineering application examples
Nonstiff flight, vibration, and control simulations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Nonstiff flight, vibration, and control simulations.
Method limitations
Explicit and not suited to strongly stiff problems without very small steps; no embedded error estimate.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses two related formulas to estimate local error and adapt the step.
Advance in timeNumerical technique
Formulation & short derivation
Representative numerical formulation
yn+1(p)=yn+hi∑bikie=hi∑(bi−bi)ki
Derivation / construction sketch
Reuse a shared set of stage slopes in two formulas of different order.
Compare their updates to estimate local error.
Accept or reject and adjust the step using scaled tolerances.
Symbols & assumptions
Representative embedded-pair form; RK45 commonly uses a Dormand-Prince 5(4) pair. Local control is not a global error guarantee.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Adaptive nonstiff ODE simulation.
Method limitations
Representative embedded-pair form; RK45 commonly uses a Dormand-Prince 5(4) pair. Local control is not a global error guarantee.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Use several past states to approximate the new-time derivative.
Advance in timeNumerical technique
Formulation & short derivation
Representative numerical formulation
2h3yn+1−4yn+yn−1=f(tn+1,yn+1)
Derivation / construction sketch
Interpolate recent states with a polynomial.
Differentiate that polynomial at the newest time.
Solve the implicit residual for the new state.
Symbols & assumptions
Constant-step BDF2 shown; higher-order formulas have different stability limits and require startup/history handling.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Chemical kinetics, batteries, and other stiff systems.
Engineering application examples
Chemical kinetics, batteries, and other stiff systems.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Chemical kinetics, batteries, and other stiff systems.
Method limitations
Constant-step BDF2 shown; higher-order formulas have different stability limits and require startup/history handling.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Drift the position with the intermediate velocity.
Apply the remaining half kick using the new force.
Symbols & assumptions
Second order and symplectic for suitable separable Hamiltonian systems at fixed step; arbitrary damping changes these properties.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Molecular dynamics and long-time particle mechanics.
Engineering application examples
Molecular dynamics and long-time particle mechanics.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Molecular dynamics and long-time particle mechanics.
Method limitations
Second order and symplectic for suitable separable Hamiltonian systems at fixed step; arbitrary damping changes these properties.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Obtains equilibrium shear viscosity from the time integral of microscopic shear-stress fluctuations.
Generate equilibrium MD trajectories and shear-pressure samples with a compatible ensemble and force field; this time integrator alone does not estimate viscosity.
Computes phonon modes from a quadratic expansion of crystal potential energy.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For compatible position-dependent conservative forces; constraints, thermostats, and stochastic forces require specialized extensions.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Treats stiff terms implicitly and nonstiff terms explicitly.
Advance in timeNumerical technique
Formulation & short derivation
Representative numerical formulation
yn+1=yn+hfE(yn)+hfI(yn+1)
Derivation / construction sketch
Split the dynamics by stiffness or computational structure.
Use an explicit update for one part and an implicit update for the other.
Combine with compatible order and stability conditions.
Symbols & assumptions
First-order IMEX Euler shown; splitting and stability constraints remain problem dependent.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Time integration
Practical use
Reaction-transport and advection-diffusion systems.
Engineering application examples
Reaction-transport and advection-diffusion systems.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Reaction-transport and advection-diffusion systems.
Method limitations
First-order IMEX Euler shown; splitting and stability constraints remain problem dependent.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Moves downhill along the negative objective gradient.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
xk+1=xk−αk∇f(xk)
Derivation / construction sketch
Use a first-order local objective approximation.
Choose a descent direction opposite to the gradient.
Select a step using a line search or a justified fixed rule.
Symbols & assumptions
Convergence depends on smoothness and step size; nonconvex objectives may have local minima.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Parameter tuning and inverse-model fitting.
Method limitations
Convergence depends on smoothness and step size; nonconvex objectives may have local minima.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses curvature to compute a local stationary-point correction.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
∇2f(xk)pk=−∇f(xk)xk+1=xk+αkpk
Derivation / construction sketch
Build a second-order Taylor model of the objective.
Set its gradient to zero.
Globalize the step with line search or a trust region.
Symbols & assumptions
An indefinite Hessian may not produce descent; regularization or trust-region safeguards may be needed.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Smooth engineering design optimization.
Method limitations
An indefinite Hessian may not produce descent; regularization or trust-region safeguards may be needed.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Updates an inverse-Hessian approximation using gradient differences.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
Hk+1=(I−ρsyT)Hk(I−ρysT)+ρssTρ=yTs1
Derivation / construction sketch
Take a descent step and measure the gradient change.
Enforce a secant relation while preserving symmetry.
Use the BFGS update and a curvature-respecting line search.
Symbols & assumptions
Positive definiteness needs positive curvature y dot s; noisy gradients can be problematic.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Optimization & inverse problems
Practical use
Smooth unconstrained calibration and shape optimization.
Engineering application examples
Smooth unconstrained calibration and shape optimization.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Smooth unconstrained calibration and shape optimization.
Method limitations
Positive definiteness needs positive curvature y dot s; noisy gradients can be problematic.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses limited curvature history with bound constraints.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
xminf(x)li≤xi≤uipk≈−Hk∇f(xk)
Derivation / construction sketch
Store a limited set of displacement and gradient-difference pairs.
Identify bound-active variables and a feasible search direction.
Apply a line search within bounds.
Symbols & assumptions
Schematic direction shown; the algorithm includes generalized Cauchy and subspace steps. Bounds do not make a nonconvex problem globally solvable.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Optimization & inverse problems
Practical use
Large parameter estimation with physical parameter bounds.
Engineering application examples
Large parameter estimation with physical parameter bounds.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large parameter estimation with physical parameter bounds.
Method limitations
Schematic direction shown; the algorithm includes generalized Cauchy and subspace steps. Bounds do not make a nonconvex problem globally solvable.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Linearizes residuals to solve a nonlinear least-squares problem.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
xmin21∥r(x)∥2JTJp=−JTr
Derivation / construction sketch
Expand each residual to first order.
Minimize the squared norm of the linearized residual.
Update and repeat, preferably solving the least-squares system by stable factorization.
Symbols & assumptions
Normal equations show the concept but square the condition number; QR/SVD can be preferable. Best near a suitable small-residual solution.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Optimization & inverse problems
Practical use
Calibration of sensors and constitutive parameters.
Engineering application examples
Calibration of sensors and constitutive parameters.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Calibration of sensors and constitutive parameters.
Method limitations
Normal equations show the concept but square the condition number; QR/SVD can be preferable. Best near a suitable small-residual solution.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Regularizes a Gauss-Newton step to balance stability and progress.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
(JTJ+λI)p=−JTr
Derivation / construction sketch
Construct the linearized residual objective.
Add a penalty on the correction size.
Adjust damping according to agreement between predicted and actual reduction.
Symbols & assumptions
Identity-scaled schematic form; practical implementations use scaling and trust-region logic.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Optimization & inverse problems
Practical use
Nonlinear curve fitting and experimental model calibration.
Engineering application examples
Nonlinear curve fitting and experimental model calibration.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Nonlinear curve fitting and experimental model calibration.
Method limitations
Identity-scaled schematic form; practical implementations use scaling and trust-region logic.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Treats crystal vibrations as independent quantum oscillators at a single frequency.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Approximates acoustic phonons by a continuum spectrum with a mode-count cutoff.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Describes conduction electrons as a degenerate, noninteracting Fermi gas.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solves a sequence of locally quadratic constrained subproblems.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
pmin21pTBkp+∇fkTpck+Jkp=0
Derivation / construction sketch
Approximate the Lagrangian curvature.
Linearize constraints and solve a quadratic subproblem.
Globalize and update primal and multiplier estimates.
Symbols & assumptions
Equality form shown; inequality constraints require active-set or related treatment and constraint qualifications.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Optimization & inverse problems
Practical use
Constrained geometry and operating-point optimization.
Engineering application examples
Constrained geometry and operating-point optimization.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Constrained geometry and operating-point optimization.
Method limitations
Equality form shown; inequality constraints require active-set or related treatment and constraint qualifications.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approaches inequality-constrained solutions through barrier subproblems.
OptimizeNumerical technique
Formulation & short derivation
Representative numerical formulation
xminf(x)−μi∑lnsi(x)si(x)>0
Derivation / construction sketch
Represent inequalities by positive slack functions.
Penalize approach to the feasible boundary with a logarithmic barrier.
Solve a sequence of primal-dual or barrier systems as the parameter decreases.
Symbols & assumptions
Feasibility, scaling, and conditioning near active constraints need care; nonconvex problems lack a general global guarantee.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Large constrained engineering design problems.
Method limitations
Feasibility, scaling, and conditioning near active constraints need care; nonconvex problems lack a general global guarantee.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Passes a polynomial through prescribed distinct data points.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
p(x)=i∑yiℓi(x)ℓi(x)=j=i∏xi−xjx−xj
Derivation / construction sketch
Construct a basis that is one at its own node and zero at all others.
Weight each basis function by its data value.
Sum to reproduce the nodal data.
Symbols & assumptions
High-degree equispaced interpolation can oscillate badly; noisy data may need fitting instead.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Tabulated material-property interpolation.
Method limitations
High-degree equispaced interpolation can oscillate badly; noisy data may need fitting instead.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Cancel the common factor between numerator and denominator.
At a node, return the stored nodal value directly.
Symbols & assumptions
A stable evaluation form cannot eliminate poor node choice or intrinsic interpolation conditioning.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Repeated evaluations of polynomial surrogate tables.
Engineering application examples
Repeated evaluations of polynomial surrogate tables.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Repeated evaluations of polynomial surrogate tables.
Method limitations
A stable evaluation form cannot eliminate poor node choice or intrinsic interpolation conditioning.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Joins piecewise cubic polynomials with continuity constraints.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
Si(x)=ai+biξ+ciξ2+diξ3S,S′,S′′ are continuous
Derivation / construction sketch
Fit a cubic on each interval.
Match values and first two derivatives at interior knots.
Add endpoint conditions to close the system.
Symbols & assumptions
Endpoint choices affect the result; ordinary cubic splines need not preserve positivity or monotonicity.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Smooth material curves and measured trajectory reconstruction.
Engineering application examples
Smooth material curves and measured trajectory reconstruction.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Smooth material curves and measured trajectory reconstruction.
Method limitations
Endpoint choices affect the result; ordinary cubic splines need not preserve positivity or monotonicity.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Treats crystal vibrations as independent quantum oscillators at a single frequency.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For smooth interpolation of coefficients or responses; ordinary splines do not guarantee positivity or monotonicity.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Fits basis coefficients by minimizing data residuals.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
c∗=argcmin∥Ac−y∥22Aij=ϕj(xi)
Derivation / construction sketch
Choose a polynomial basis and assemble a design matrix.
Project the data onto its column space.
Use QR or SVD to obtain a stable least-squares solution.
Symbols & assumptions
Scale coordinates and avoid unnecessarily high degree; outliers and correlated errors require additional treatment.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Calibration curves and response-surface approximations.
Engineering application examples
Calibration curves and response-surface approximations.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Calibration curves and response-surface approximations.
Method limitations
Scale coordinates and avoid unnecessarily high degree; outliers and correlated errors require additional treatment.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses Chebyshev bases and clustered nodes to approximate smooth functions.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
pN(x)=k=0∑NakTk(x)Tk(cosθ)=cos(kθ)
Derivation / construction sketch
Map the domain to the standard interval.
Sample or project using Chebyshev structure.
Truncate the expansion after checking coefficient decay or error.
Symbols & assumptions
Smoothness controls convergence; discontinuities cause Gibbs-type behavior.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Accurate one-dimensional property and response surrogates.
Engineering application examples
Accurate one-dimensional property and response surrogates.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Accurate one-dimensional property and response surrogates.
Method limitations
Smoothness controls convergence; discontinuities cause Gibbs-type behavior.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Retain dominant modes and project the dynamics or data onto that subspace.
Symbols & assumptions
Optimal low-rank snapshot error does not guarantee predictive accuracy or stable reduced dynamics.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Reduced fluid, structural, and thermal models.
Method limitations
Optimal low-rank snapshot error does not guarantee predictive accuracy or stable reduced dynamics.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Fits a linear evolution map between successive snapshots.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
A≈YX+A=UrTYVrΣr−1
Derivation / construction sketch
Arrange consecutive snapshots in paired matrices.
Fit a least-squares map from present to future snapshots.
Project the map and analyze its eigenvalues and modes.
Symbols & assumptions
Sampling, noise, rank truncation, and nonlinearity affect interpretation and extrapolation.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Flow-pattern analysis and data-driven dynamics prediction.
Engineering application examples
Flow-pattern analysis and data-driven dynamics prediction.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Flow-pattern analysis and data-driven dynamics prediction.
Method limitations
Sampling, noise, rank truncation, and nonlinearity affect interpretation and extrapolation.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Predicts a response using a covariance model and observed data.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
m∗=k∗T(K+σn2I)−1yv∗=k∗∗−k∗T(K+σn2I)−1k∗
Derivation / construction sketch
Specify a prior mean and covariance kernel.
Condition the joint Gaussian distribution on observations.
Compute the posterior mean and variance at a query point.
Symbols & assumptions
Zero-mean formula shown; uncertainty is conditional on the kernel/noise assumptions, not a guarantee of physical accuracy.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Approximation & reduction
Practical use
Expensive-simulation surrogates and uncertainty-aware design exploration.
Engineering application examples
Expensive-simulation surrogates and uncertainty-aware design exploration.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Expensive-simulation surrogates and uncertainty-aware design exploration.
Method limitations
Zero-mean formula shown; uncertainty is conditional on the kernel/noise assumptions, not a guarantee of physical accuracy.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Integrates sampled data by joining neighboring values with straight lines.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
Ih=h[21f0+i=1∑n−1fi+21fn]
Derivation / construction sketch
Interpolate each interval linearly.
Integrate that line exactly.
Sum the interval contributions.
Symbols & assumptions
Uniform-spacing formula shown; second-order error for sufficiently smooth functions.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Integrating measured heat-flow, force, or current histories.
Engineering application examples
Integrating measured heat-flow, force, or current histories.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Integrating measured heat-flow, force, or current histories.
Method limitations
Uniform-spacing formula shown; second-order error for sufficiently smooth functions.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Integrates pairs of intervals using quadratic interpolation.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
Ih=3h[f0+4i odd∑fi+2i even,0<i<n∑fi+fn]
Derivation / construction sketch
Fit a quadratic across each consecutive pair of intervals.
Integrate the polynomial exactly.
Add the repeated weights across the domain.
Symbols & assumptions
Uniform grid with even n shown; fourth-order convergence requires smoothness.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Integrating smooth sampled loads and response curves.
Engineering application examples
Integrating smooth sampled loads and response curves.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Integrating smooth sampled loads and response curves.
Method limitations
Uniform grid with even n shown; fourth-order convergence requires smoothness.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Chooses nodes and weights to integrate high-degree polynomials efficiently.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
∫−11f(x)dx≈i=1∑nwif(xi)
Derivation / construction sketch
Choose nodes as roots of the relevant orthogonal polynomial.
Determine weights by exactness conditions.
Map the rule to the integration interval or element.
Symbols & assumptions
Gauss-Legendre is exact through degree 2n-1; singular or nonsmooth integrands need special treatment.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Element stiffness and load integrals in finite-element analysis.
Engineering application examples
Element stiffness and load integrals in finite-element analysis.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Element stiffness and load integrals in finite-element analysis.
Method limitations
Gauss-Legendre is exact through degree 2n-1; singular or nonsmooth integrands need special treatment.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Approximates acoustic phonons by a continuum spectrum with a mode-count cutoff.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Describes conduction electrons as a degenerate, noninteracting Fermi gas.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Builds crystal electronic bands from localized orbitals and intersite hopping.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Predicts band gaps by perturbing free electrons with a weak periodic potential.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Subdivides intervals according to local integration-error estimates.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
I≈K∑QKK∑eK≤max(εabs,εrel∣I∣)
Derivation / construction sketch
Compare paired integration rules to estimate local error.
Refine intervals with the largest estimated contribution.
Stop when the global estimated tolerance is met.
Symbols & assumptions
Estimates can miss singularities or narrow features; supply known breakpoints and examine diagnostics.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Accurate one-dimensional response and probability integrals.
Engineering application examples
Accurate one-dimensional response and probability integrals.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Accurate one-dimensional response and probability integrals.
Method limitations
Estimates can miss singularities or narrow features; supply known breakpoints and examine diagnostics.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Approximates acoustic phonons by a continuum spectrum with a mode-count cutoff.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For deterministic low-dimensional integrals; identify singularities and verify error estimates.
Describes conduction electrons as a degenerate, noninteracting Fermi gas.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For deterministic low-dimensional integrals; identify singularities and verify error estimates.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Estimates an integral by averaging independent random samples.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
IN=N1i=1∑Nf(Xi)SE(IN)≈Ns
Derivation / construction sketch
Express the target as an expectation under a chosen sampling distribution.
Generate independent samples and average the integrand.
Estimate uncertainty from sample variability.
Symbols & assumptions
Finite-variance independent-sample form; dimension-independent rate can still have a large variance constant.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Uncertain-load propagation and probabilistic engineering estimates.
Engineering application examples
Uncertain-load propagation and probabilistic engineering estimates.
Sampling building block; construct the integrand estimator and its uncertainty analysis in application code.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Uncertain-load propagation and probabilistic engineering estimates.
Method limitations
Finite-variance independent-sample form; dimension-independent rate can still have a large variance constant.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses low-discrepancy points to cover an integration domain evenly.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
I≈N1i=1∑Nf(ui)ui∈[0,1]d
Derivation / construction sketch
Map the integral to a unit cube.
Generate a low-discrepancy design such as Sobol points.
Average values; use independent randomized scramblings for error assessment.
Symbols & assumptions
Performance depends on smoothness and effective dimension; deterministic points do not give an IID standard error.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
High-dimensional uncertainty propagation and design integration.
Engineering application examples
High-dimensional uncertainty propagation and design integration.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
High-dimensional uncertainty propagation and design integration.
Method limitations
Performance depends on smoothness and effective dimension; deterministic points do not give an IID standard error.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Changes the sampling distribution to focus on influential regions.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
I=∫f(x)p(x)dx=Eq[f(X)p(X)/q(X)]
Derivation / construction sketch
Choose a proposal density covering the target contribution.
Sample from that proposal.
Weight each value by the target-to-proposal density ratio.
Symbols & assumptions
Support coverage and finite weight variance are essential; extreme weights can destroy efficiency.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Rare-event estimation and reliability calculations.
Engineering application examples
Rare-event estimation and reliability calculations.
Implementation route: use a sampling library and compute density-ratio weights; verify support and weight diagnostics. The review explains the construction and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Rare-event estimation and reliability calculations.
Method limitations
Support coverage and finite weight variance are essential; extreme weights can destroy efficiency.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Builds a Markov chain with a desired stationary density.
Approximate & sampleNumerical technique
Formulation & short derivation
Representative numerical formulation
α(x,y)=min(1,π(x)q(y∣x)π(y)q(x∣y))
Derivation / construction sketch
Propose a move using a chosen transition distribution.
Accept according to the target/proposal ratio.
Repeat and diagnose mixing and convergence.
Symbols & assumptions
Samples are correlated; initialization, ergodicity, effective sample size, and multimodality matter.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Quadrature & sampling
Practical use
Bayesian inverse problems and posterior uncertainty exploration.
Engineering application examples
Bayesian inverse problems and posterior uncertainty exploration.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Bayesian inverse problems and posterior uncertainty exploration.
Method limitations
Samples are correlated; initialization, ergodicity, effective sample size, and multimodality matter.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Cancels a leading discretization-error term using two resolutions.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
u∗≈uh/r+rp−1uh/r−uh
Derivation / construction sketch
Assume a leading error term proportional to h to the power p.
Write that expansion at two consistently refined resolutions.
Eliminate the leading coefficient.
Symbols & assumptions
Requires an asymptotic convergence regime and a valid order estimate; shocks and inconsistent grids can invalidate the assumption.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Implementation route: compare systematically refined solver runs and evaluate the error model; no particular solver is certified by the estimate.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Checking mesh and time-step convergence.
Method limitations
Requires an asymptotic convergence regime and a valid order estimate; shocks and inconsistent grids can invalidate the assumption.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Treats crystal vibrations as independent quantum oscillators at a single frequency.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For systematically refined computations in an established asymptotic error regime; use consistent geometry and boundary data.
Computes phonon modes from a quadratic expansion of crystal potential energy.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For systematically refined computations in an established asymptotic error regime; use consistent geometry and boundary data.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Reports a safety-factored estimate of discretization uncertainty.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
GCIfine=Fsrp−1∣(uf−uc)/uf∣
Derivation / construction sketch
Estimate the observed convergence order using a systematic refinement study.
Use a Richardson-style fine-grid error estimate.
Apply the chosen safety factor and report assumptions.
Symbols & assumptions
Relative form fails near zero output; nonmonotonic convergence and coupled grid/time errors need special analysis.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Implementation route: compute the index from a documented systematic refinement study; the report explains numerical verification practices.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Documenting numerical uncertainty in CFD outputs.
Method limitations
Relative form fails near zero output; nonmonotonic convergence and coupled grid/time errors need special analysis.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tests a PDE implementation using a constructed exact solution.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
Lu=fu=um⇒fm=Lum
Derivation / construction sketch
Choose a smooth analytic manufactured solution.
Substitute it into the governing operator to compute forcing and boundary data.
Solve and verify that error decreases at the expected rate.
Symbols & assumptions
Tests implementation accuracy, not whether the governing physical model describes reality.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Error analysis & verification
Practical use
Code verification for PDE and multiphysics solvers.
Engineering application examples
Code verification for PDE and multiphysics solvers.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Code verification for PDE and multiphysics solvers.
Method limitations
Tests implementation accuracy, not whether the governing physical model describes reality.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses equation and interface residuals to guide error assessment.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
ηK2=hK2∥f+∇⋅(k∇uh)∥K2+e⊂∂K∑he∥Je∥e2
Derivation / construction sketch
Measure how strongly the approximate solution violates the PDE inside cells.
Add flux-jump contributions on interfaces.
Combine with problem-dependent weights and refine where indicators are large.
Symbols & assumptions
Schematic elliptic estimator; reliability constants and boundary terms depend on assumptions and discretization.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
MFEM supports estimator-driven adaptivity; the displayed residual form is schematic and is not a claim about the exact estimator selected by an example.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Adaptive refinement in finite-element simulation.
Method limitations
Schematic elliptic estimator; reliability constants and boundary terms depend on assumptions and discretization.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tests Fourier-mode amplification for linear grid schemes.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
ujn=Gneijθ∣G(θ)∣≤1
Derivation / construction sketch
Insert a Fourier mode into a linear constant-coefficient difference scheme.
Solve for its amplification factor.
Require bounded amplification for every resolvable wave number.
Symbols & assumptions
Typically periodic or infinite uniform grids; boundary effects and nonlinear stability need separate analysis.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Error analysis & verification
Practical use
Selecting stable explicit diffusion and wave time steps.
Engineering application examples
Selecting stable explicit diffusion and wave time steps.
Implementation route: derive the amplification factor and evaluate its magnitude over wave numbers; this is an analysis procedure rather than a solver product.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Selecting stable explicit diffusion and wave time steps.
Method limitations
Typically periodic or infinite uniform grids; boundary effects and nonlinear stability need separate analysis.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Measures how perturbations in inputs can affect a computed solution.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
κ(A)=∥A∥∥A−1∥∥x∥∥δx∥≤κ(A)∥b∥∥δb∥
Derivation / construction sketch
Perturb a nonsingular linear system with fixed A.
Bound the solution change using operator norms.
Compare relative input and output perturbations.
Symbols & assumptions
The bound shown concerns right-hand-side perturbations; conditioning is a problem property, distinct from algorithm stability.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Error analysis & verification
Practical use
Diagnosing sensitive inverse problems and poorly scaled matrix systems.
Engineering application examples
Diagnosing sensitive inverse problems and poorly scaled matrix systems.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Diagnosing sensitive inverse problems and poorly scaled matrix systems.
Method limitations
The bound shown concerns right-hand-side perturbations; conditioning is a problem property, distinct from algorithm stability.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Builds crystal electronic bands from localized orbitals and intersite hopping.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For assembled linear systems or fitting matrices; separate problem conditioning from algorithm stability.
Predicts band gaps by perturbing free electrons with a weak periodic potential.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For assembled linear systems or fitting matrices; separate problem conditioning from algorithm stability.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Estimates an analytic derivative without real subtractive cancellation.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
f′(x)≈hImf(x+ih)
Derivation / construction sketch
Expand an analytic function in a complex Taylor series.
Extract the imaginary part.
Divide by the perturbation to recover the derivative with second-order truncation error.
Symbols & assumptions
Requires holomorphic operations and complex-compatible code; absolute values, branches, or discarded imaginary parts can break it.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Error analysis & verification
Practical use
Checking gradients in smooth engineering optimization code.
Engineering application examples
Checking gradients in smooth engineering optimization code.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Checking gradients in smooth engineering optimization code.
Method limitations
Requires holomorphic operations and complex-compatible code; absolute values, branches, or discarded imaginary parts can break it.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Computes gradients of scalar outputs with respect to many parameters.
Verify & assessVerification / analysis
Formulation & short derivation
Representative numerical formulation
R(u,p)=0RuTλ=JuTdpdJ=Jp−λTRp
Derivation / construction sketch
Differentiate the state residual and objective.
Introduce an adjoint variable to eliminate the expensive state sensitivity.
Solve one adjoint system per scalar objective and assemble parameter gradients.
Symbols & assumptions
Discrete formulation shown; consistent boundary conditions, differentiation, and solver tolerances are essential.
The sketch summarizes algorithm construction, not a complete convergence proof or implementation. Consult the references for stopping criteria, stability requirements, and variants.
Practical applications & products
Application area
Error analysis & verification
Practical use
Aerodynamic shape optimization and inverse design.
Engineering application examples
Aerodynamic shape optimization and inverse design.
Named software documentation describes this method or a directly relevant implementation component; verify the selected routine and its assumptions.
Application examples illustrate where a technique is useful. They do not assert an undocumented manufacturer workflow or certify a software implementation.
Example, limitations & references
In practice
Aerodynamic shape optimization and inverse design.
Method limitations
Discrete formulation shown; consistent boundary conditions, differentiation, and solver tolerances are essential.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Choose the method that fits your equations and error budget.
01
Define the outcome
Choose the quantities, length and time scales, operating conditions, and accuracy you need. Distinguish discretization error, algebraic error, roundoff, and uncertainty in the physical inputs.
02
Check the assumptions
Check matrix structure, stiffness, smoothness, constraints, conservation, and available derivative information before selecting a numerical method.
03
Verify, then validate
Check units, conservation, numerical convergence and limiting cases. Compare predictions with independent measurements and report uncertainty and model limitations.
ABOUT THIS ATLAS
A starting point for discovery.
Numerical Modeling brings together discretizations, solvers, approximations, sampling techniques, and verification tools. A numerical method approximates or solves a mathematical problem; it does not establish whether the underlying physical model describes reality.
This release includes 68 entries across 8 subject areas. Methods may have many variants, and software implementations can differ. A converged iteration is not by itself proof of discretization accuracy or physical validity.
References at the point of use
Open any entry’s “Example, limitations & references” section for linked papers, author-written textbooks, or official technical documentation. A technical manual can support several related entries; use the model name to find the relevant section. Some publisher-hosted papers require access.
Descriptions and examples are concise editorial summaries. This educational catalog is not a simulation service, engineering certification, or substitute for validating a design. Follow the licensing and citation requirements of each original source.
Catalog release: September 23, 2026 · Version 1.0
THE NUMERICAL MODELING COMMUNITY
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Numerical Modeling is an educational catalog of numerical techniques, algorithms, and verification methods. Its descriptions, equations, derivation sketches, relationships, and examples provide general information. They are not professional engineering, scientific, medical, legal, or other advice, design approval, certification, or a guarantee of safety or performance.
Limitations and independent verification
Formulations are representative and simplified. Assumptions, boundary conditions, parameter ranges, units, and implementation details can change a model’s validity. The catalog and relationship tree are not exhaustive; entries may contain errors, omissions, or outdated information. Software and application examples do not establish suitability for a particular application. Check current primary references, independently verify calculations, validate against appropriate evidence, and obtain review by qualified professionals before practical use. Do not rely on this site alone for safety-critical design, operation, or decisions.
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External sources and third-party products
References and search links lead to independently operated sites with their own terms and policies. Listing a source, company, product, or implementation does not imply affiliation, endorsement, or a verified manufacturer workflow. Third-party names, trademarks, software, publications, and other materials remain subject to their owners’ rights and applicable licenses. A link or citation does not grant permission to reuse those materials.