1D Euler Sod#

This problem solves the 1D conservative Euler equations for the Sod1d problem.

\[\begin{split}\frac{\partial }{\partial t} \begin{bmatrix}\rho \\ \rho u\\ \rho E \end{bmatrix} + \frac{\partial }{\partial x} \begin{bmatrix}\rho u \\ \rho u^2 +p\\ u(E+p) \end{bmatrix} = 0\end{split}\]

where the pressure \(p\) is related to the conserved quantities through the equation of the state

\[p=(\gamma -1)(\rho E-\frac{1}{2}\rho u^2).\]

  • Initial conditions in primitive variables:

    • \(x<=0 :\quad \rho =1, u = 0, p = 1\)

    • \(x>0 :\quad \rho =1/8, u = 0, p = 0.1\)

    • These are used to create the initial conditions in conservative variables.

  • By default, \(\gamma = 1.4\)

  • Domain is \([-0.5, 0.5]\) with homogeneous Neumann BC

  • Typically, integration is performed over \(t \in (0, 0.2)\)

Mesh#

python3 pressio-demoapps/meshing_scripts/create_full_mesh_for.py \
        --problem sod1d_s<stencilSize> -n <N> --outDir <destination-path>

where:

  • N is the number of cells you want

  • <stencilSize> = 3 or 5 or 7: defines the neighboring connectivity of each cell

  • <destination-path>: full path to where you want the mesh files to be generated. The script creates the directory if it does not exist.

Important

When you set the <stencilSize>, keep in mind the following constraints (more on this below):

  • InviscidFluxReconstruction::FirstOrder requires <stencilSize> >= 3

  • InviscidFluxReconstruction::Weno3 requires <stencilSize> >= 5

  • InviscidFluxReconstruction::Weno5 requires <stencilSize> >= 7

C++ synopsis#

#include "pressiodemoapps/euler1d.hpp"

int main(){
  namespace pda = pressiodemoapps;

  const auto meshObj = pda::load_cellcentered_uniform_mesh_eigen("path-to-mesh");

  const auto probId = pda::Euler1d::Sod;
  const auto scheme = pda::InviscidFluxReconstruction::FirstOrder; //or Weno3, Weno5
  auto problem      = pda::create_problem_eigen(meshObj, probId, scheme);
}

Python synopsis#

import pressiodemoapps as pda

meshObj = pda.load_cellcentered_uniform_mesh("path-to-mesh")

probId  = pda.Euler1d.Sod
scheme  = pda.InviscidFluxReconstruction.FirstOrder # or Weno3, Weno5
problem = pda.create_problem(meshObj, probId, scheme)

Sample Solution#

Representative plot for total simulation time \(T=0.2\) showing density at selected time steps \(t \in \left \{0, 0.1, 0.2\right \}\) obtained using time step \(dt = 10^{-4}\), Weno5, Runge-Kutta4 integration with a mesh of \(N=1000\) cells.

Alternative text