m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Shock capturing in solids and fluids

Derive conservative finite-volume updates, jump conditions, stable fluxes and the distinctions between fluid shocks and solid stress waves.

Subject library · 51 guides · derivations & worked examples

Fundamental physics

Space and flight

Materials and transport

Systems and mathematics

Marine and road vehicles

Electrical and computing

Companion site

Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Conservation and discontinuities

Definitions & inputs. U conserved state,F physical flux,s discontinuity speed,[a]=aR−aL.

  1. Local conservation is the starting PDE.

    ∂tU+∂xF(U)=0\partial_tU+\partial_xF(U)=0

  2. Integrate over a fixed control volume.

    ddt∫xLxRU dx=F(U(xL))−F(U(xR))\frac{d}{dt}\int_{x_L}^{x_R}U\,dx=F(U(x_L))-F(U(x_R))

  3. Integrate across a moving thin discontinuity to obtain Rankine–Hugoniot.

    s[U]=[F]s[U]=[F]

Interpretation. An entropy condition selects physically admissible compressive shocks rather than arbitrary weak solutions.

↑ Return to definitions and contents

2. Finite volume and numerical flux

Definitions & inputs. Uin cell average,Δx cell width,Δt step,Fhat interface flux,a maximum wave speed.

  1. Neighboring cells share one numerical interface flux so internal transfers cancel.

    Uin+1=Uin−ΔtΔx(F^i+1/2−F^i−1/2)U_i^{n+1}=U_i^n-\frac{\Delta t}{\Delta x}(\widehat F_{i+1/2}-\widehat F_{i-1/2})

  2. Rusanov flux adds dissipation based on a bound on characteristic speeds.

    F^=12(FL+FR)−a2(UR−UL)\widehat F=\tfrac12(F_L+F_R)-\tfrac a2(U_R-U_L)

  3. Choose a stable CFL number for the method and dimension; a smaller time step does not compensate for wrong physics.

    Δt≤CCFLΔx/amax\Delta t\le C_{CFL}\Delta x/a_{max}

Interpretation. Higher-order reconstruction needs limiting near shocks; unlimited interpolation can produce oscillations and negative states.

↑ Return to definitions and contents

3. Compressible fluid waves

Definitions & inputs. U=(ρ,ρu,ρE),E=e+u²/2,p pressure,γ heat-capacity ratio.

  1. Conserve mass, momentum and total energy together.

    F=(ρu,ρu2+p,u(ρE+p))TF=(\rho u,\rho u^2+p,u(\rho E+p))^{\mathsf T}

  2. Close the system with the ideal-gas equation of state.

    p=(γ−1)ρe,c=γp/ρp=(\gamma-1)\rho e,\quad c=\sqrt{\gamma p/\rho}

  3. Acoustic waves flank the material contact; HLLC restores a contact omitted by a two-wave HLL approximation.

    λ=u−c, u, u+c\lambda=u-c,\ u,\ u+c

Interpretation. Roe, HLL/HLLC, approximate Riemann solvers, MUSCL, WENO and discontinuous Galerkin require positivity, entropy and boundary checks. Use shock tubes and smooth convergence tests.

↑ Return to definitions and contents

4. Solid stress waves and verification

Definitions & inputs. ε strain,v particle velocity,σ tensile-positive stress,E modulus,ρ reference density.

  1. Compatibility and momentum form a hyperbolic system.

    ∂tϵ−∂xv=0,∂t(ρv)−∂xσ=0\partial_t\epsilon-\partial_xv=0,\quad\partial_t(\rho v)-\partial_x\sigma=0

  2. A slender bar supports longitudinal waves with mechanical impedance Z; bulk 3D wave speeds use different moduli.

    σ=Eϵ⇒cL=E/ρ,Z=ρcL\sigma=E\epsilon\Rightarrow c_L=\sqrt{E/\rho},\quad Z=\rho c_L

  3. Stress reflection follows continuity of velocity and traction at an ideal bonded interface.

    Rσ=(Z2−Z1)/(Z2+Z1)R_\sigma=(Z_2-Z_1)/(Z_2+Z_1)

Interpretation. For strong solid shocks evolve energy, deformation and plastic internal variables consistently. Check wave speed, interface reflection, jump residuals, mass/energy balance and grid convergence; artificial viscosity is numerical regularization, not physical viscosity.

↑ Return to definitions and contents

Graphical worked example

Ideal-gas normal-shock benchmark with γ=1.4. Constant-gamma limitations apply at high enthalpy. X axis: Upstream normal Mach number (dimensionless). Y axis: Normal-shock pressure ratio (dimensionless).
Ideal-gas normal-shock benchmark with γ=1.4. Constant-gamma limitations apply at high enthalpy. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Burgers shock speed

Definitions & inputs. f(u)=u²/2,uL2,uR0 in consistent normalized units.

  1. Choose the governing model and isolate the requested quantity.

    s=(fR−fL)/(uR−uL)s=(f_R-f_L)/(u_R-u_L)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (0−2)/(0−2)(0-2)/(0-2)

  3. Evaluate the expression; the result uses the units shown.

    Result=1 \mathrm{Result}=1\ {}

Interpretation. Compressive shock withuL>uR.

↑ Return to definitions and contents
Example 02. Shock position

Definitions & inputs. Initial x0=0,s1,t.4.

  1. Choose the governing model and isolate the requested quantity.

    x=x0+stx=x_0+st

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    0+1(.4)0+1(.4)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4 \mathrm{Result}=0.4\ {}

Interpretation. Constant-speed discontinuity.

↑ Return to definitions and contents
Example 03. Acoustic speed

Definitions & inputs. γ1.4,p100000Pa,ρ1.2kg/m³.

  1. Choose the governing model and isolate the requested quantity.

    c=γp/ρc=\sqrt{\gamma p/\rho}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.4(100000)/1.2\sqrt{1.4(100000)/1.2}

  3. Evaluate the expression; the result uses the units shown.

    Result=341.565 m/s\mathrm{Result}=341.565\ \mathrm{m/s}

Interpretation. Ideal gas.

↑ Return to definitions and contents
Example 04. Maximum Euler speed

Definitions & inputs. u100m/s,c340m/s.

  1. Choose the governing model and isolate the requested quantity.

    a=∣u∣+ca=|u|+c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    100+340100+340

  3. Evaluate the expression; the result uses the units shown.

    Result=440 m/s\mathrm{Result}=440\ \mathrm{m/s}

Interpretation. Bounds one-dimensional characteristics.

↑ Return to definitions and contents
Example 05. CFL time step

Definitions & inputs. Δx.01m,CFL.5,a500m/s.

  1. Choose the governing model and isolate the requested quantity.

    Δt=CΔx/a\Delta t=C\Delta x/a

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(.01)/500.5(.01)/500

  3. Evaluate the expression; the result uses the units shown.

    Result=1×10−5 s\mathrm{Result}=1\times10^{-5}\ \mathrm s

Interpretation. Explicit method.

↑ Return to definitions and contents
Example 06. Courant number

Definitions & inputs. a2m/s,Δt.1s,Δx.5m.

  1. Choose the governing model and isolate the requested quantity.

    C=aΔt/ΔxC=a\Delta t/\Delta x

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2(.1)/.52(.1)/.5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4 \mathrm{Result}=0.4\ {}

Interpretation. Compare with method-specific limit.

↑ Return to definitions and contents
Example 07. Conservative update

Definitions & inputs. U1,dt/dx.1,right flux3,left flux2.

  1. Choose the governing model and isolate the requested quantity.

    Un+1=Un−λ(FR−FL)U^{n+1}=U^n-\lambda(F_R-F_L)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−.1(3−2)1-.1(3-2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9 \mathrm{Result}=0.9\ {}

Interpretation. Normalized scalar example.

↑ Return to definitions and contents
Example 08. Upwind flux

Definitions & inputs. Positive advection a2,uL3.

  1. Choose the governing model and isolate the requested quantity.

    F^=auL\widehat F=au_L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2(3)2(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=6 \mathrm{Result}=6\ {}

Interpretation. Exact scalar upwind flux.

↑ Return to definitions and contents
Example 09. Rusanov scalar flux

Definitions & inputs. Advection a2,uL3,uR1.

  1. Choose the governing model and isolate the requested quantity.

    F^=(2uL+2uR)/2−2(uR−uL)/2\widehat F=(2u_L+2u_R)/2-2(u_R-u_L)/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (6+2)/2−(1−3)(6+2)/2-(1-3)

  3. Evaluate the expression; the result uses the units shown.

    Result=6 \mathrm{Result}=6\ {}

Interpretation. Reduces to upwind with exact speed bound.

↑ Return to definitions and contents
Example 10. Minmod slope

Definitions & inputs. Candidate slopes2and−1.

  1. Choose the governing model and isolate the requested quantity.

    minmod⁡(2,−1)=0\operatorname{minmod}(2,-1)=0

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    00

  3. Evaluate the expression; the result uses the units shown.

    Result=0 \mathrm{Result}=0\ {}

Interpretation. Opposite signs flatten a local extremum.

↑ Return to definitions and contents
Example 11. Minmod same signs

Definitions & inputs. Candidate slopes2and1.

  1. Choose the governing model and isolate the requested quantity.

    minmod⁡(2,1)=1\operatorname{minmod}(2,1)=1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    11

  3. Evaluate the expression; the result uses the units shown.

    Result=1 \mathrm{Result}=1\ {}

Interpretation. Choose smaller magnitude.

↑ Return to definitions and contents
Example 12. Pressure from energy

Definitions & inputs. γ1.4,ρE300000J/m³,ρ1,u100m/s.

  1. Choose the governing model and isolate the requested quantity.

    p=(γ−1)(ρE−ρu2/2)p=(\gamma-1)(\rho E-\rho u^2/2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .4(300000−1002/2).4(300000-100^2/2)

  3. Evaluate the expression; the result uses the units shown.

    Result=118000 Pa\mathrm{Result}=118000\ \mathrm{Pa}

Interpretation. Total energy includes kinetic energy.

↑ Return to definitions and contents
Example 13. Normal shock density ratio

Definitions & inputs. M1=2,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    ρ2/ρ1=(γ+1)M12/[2+(γ−1)M12]\rho_2/\rho_1=(\gamma+1)M_1^2/[2+(\gamma-1)M_1^2]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2.4(4)/(2+.4(4))2.4(4)/(2+.4(4))

  3. Evaluate the expression; the result uses the units shown.

    Result=2.666667 \mathrm{Result}=2.666667\ {}

Interpretation. Ideal-gas stationary normal shock.

↑ Return to definitions and contents
Example 14. Normal shock pressure ratio

Definitions & inputs. Same upstream.

  1. Choose the governing model and isolate the requested quantity.

    p2/p1=1+2γ(M12−1)/(γ+1)p_2/p_1=1+2\gamma(M_1^2-1)/(\gamma+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+2.8(3)/2.41+2.8(3)/2.4

  3. Evaluate the expression; the result uses the units shown.

    Result=4.5 \mathrm{Result}=4.5\ {}

Interpretation. Irreversible pressure rise.

↑ Return to definitions and contents
Example 15. Bar wave speed

Definitions & inputs. E200GPa,ρ7800kg/m³.

  1. Choose the governing model and isolate the requested quantity.

    c=E/ρc=\sqrt{E/\rho}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    200×109/7800\sqrt{200\times10^9/7800}

  3. Evaluate the expression; the result uses the units shown.

    Result=5063.697 m/s\mathrm{Result}=5063.697\ \mathrm{m/s}

Interpretation. Slender elastic bar.

↑ Return to definitions and contents
Example 16. Bar transit time

Definitions & inputs. L1m,c5000m/s.

  1. Choose the governing model and isolate the requested quantity.

    t=L/ct=L/c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/50001/5000

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0002 s\mathrm{Result}=0.0002\ \mathrm s

Interpretation. One-way transit.

↑ Return to definitions and contents
Example 17. Impedance

Definitions & inputs. ρ7800kg/m³,c5000m/s.

  1. Choose the governing model and isolate the requested quantity.

    Z=ρcZ=\rho c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    7800(5000)7800(5000)

  3. Evaluate the expression; the result uses the units shown.

    Result=3.9×107 Pa s/m\mathrm{Result}=3.9\times10^{7}\ \mathrm{Pa\,s/m}

Interpretation. Longitudinal mechanical impedance.

↑ Return to definitions and contents
Example 18. Stress reflection

Definitions & inputs. Z2=2Z1.

  1. Choose the governing model and isolate the requested quantity.

    R=(Z2−Z1)/(Z2+Z1)R=(Z_2-Z_1)/(Z_2+Z_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (2−1)/(2+1)(2-1)/(2+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3333333 \mathrm{Result}=0.3333333\ {}

Interpretation. Positive reflected stress at higher impedance.

↑ Return to definitions and contents
Example 19. Matched interface

Definitions & inputs. Z2=Z1.

  1. Choose the governing model and isolate the requested quantity.

    R=(Z2−Z1)/(Z2+Z1)R=(Z_2-Z_1)/(Z_2+Z_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (1−1)/(1+1)(1-1)/(1+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0 \mathrm{Result}=0\ {}

Interpretation. No reflection in this model.

↑ Return to definitions and contents
Example 20. Observed order

Definitions & inputs. Errors.04onh,.01onh/2.

  1. Choose the governing model and isolate the requested quantity.

    p=ln⁡(eh/eh/2)/ln⁡2p=\ln(e_h/e_{h/2})/\ln2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ln⁡4/ln⁡2\ln4/\ln2

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. Smooth-problem rate; shocks often reduce global convergence order.

↑ Return to definitions and contents

Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.