Worked Physics-Model Examples

This page maps common PDEs to Exasim model choices and callback functions. The examples are formulation guides, not complete application listings.

1. Linear Advection

\[ \partial_t u + a \cdot \nabla u = 0. \]

Item	Mapping
Model type	ModelC
u	Scalar transported quantity.
q	Not present.
w	Usually absent.
v	Optional prescribed velocity field.
Flux	flux: \(F = a u\).
Source	source: zero or forcing.
Boundary/IC	ubou, fbou, fbouhdg, initu.
Examples	examples/Advection/GaussianRotating, examples/Advection/GaussianTransport.

2. Compressible Euler

Item	Mapping
Model type	ModelC
u	Conservative density, momentum, and energy.
q	Not present.
EOS	Pressure, temperature, sound speed from eos or helper functions.
Flux	Inviscid Euler flux.
Source	Body force or reaction source if present.
AV	Optional for shocks.
Examples	examples/Euler/EulerVortex, examples/Euler/naca0012, examples/Euler/mach8cylinder.

3. Heat Equation

\[ \partial_t u - \nabla \cdot (\kappa \nabla u) = s. \]

Item	Mapping
Model type	ModelD
u	Temperature or scalar field.
q	Gradient variable with \(q + \nabla u = 0\).
Flux	Conductive flux written using q, for example \(F = \kappa q\).
Source	Heat source.
Examples	examples/HeatEquation/warmup, examples/HeatEquation/cooldown.

4. Advection-Diffusion

Item	Mapping
Model type	ModelD
u	Concentration or scalar transported field.
q	Gradient for diffusive flux.
Flux	Convective plus diffusive flux.
Source	Reaction or forcing.
Examples	examples/ConvectionDiffusion/1D, examples/ConvectionDiffusion/BoundaryLayer.

5. Compressible Navier-Stokes

Item	Mapping
Model type	ModelD
u	Conservative flow state.
q	Gradient block used for viscous fluxes.
EOS	Pressure, temperature, sound speed, transport coefficients.
Flux	Inviscid plus viscous flux.
Source	Body force, chemistry, or axisymmetric terms if present.
AV	Optional for shocks/hypersonic examples.
Examples	apps/navierstokes/naca0012steady, examples/NavierStokes/flatplate2d.

6. Acoustic Wave Equation

Item	Mapping
Model type	ModelW
u	Pressure-like or displacement-like state.
q	Time-evolved wave auxiliary state.
Flux/source	First-order wave-system terms.
Initial data	initu, initq, and initw in current wave examples.
Examples	examples/WaveEquation/Membrane, examples/WaveEquation/Scattering.

7. Reactive Flow With Auxiliary ODEs

Item	Mapping
Model type	ModelC or ModelD, depending on viscosity/diffusion.
u	Flow conservative variables.
q	Present for viscous/diffusive terms.
w	Chemistry or nonequilibrium auxiliary state.
EOS	Depends on u and w.
Source	Flow source plus sourcew for auxiliary state.
Examples	examples/Euler/reacting_flow, examples/ReactingFlows/reactingcylinder.

8. Shock-Capturing Compressible Flow With AV

Item	Mapping
Model type	ModelC for inviscid, ModelD for viscous.
AV	AV flags, avfield, and/or AV data in v.
Flux	Physical flux augmented by AV contributions.
Examples	examples/Euler/mach8cylinder, examples/NavierStokes/flaredplate2d.

9. Multiphysics Example

Item	Mapping
Model type	Usually ModelD for coupled transport/diffusion, or ModelC for first-order systems.
u	Packed coupled primary state.
w	Auxiliary chemistry, thermal, or material state.
v	Prescribed or lagged coupled fields.
EOS	Shared closure across fields.
Examples	examples/GSI/*, examples/MHD/MagneticVortex, high-temperature gas modeling code.

10. Multi-Model Example

Item	Mapping
Model type	Multiple model definitions or a monolithic packed model.
Coupling	Source, boundary/interface, external-field, or auxiliary-state coupling.
Data ownership	Each exchanged field should have one owner and documented consumers.
Implementation	Use modelnumber, multiple model definitions, v, w, Fint/Fext, or source terms as appropriate.