2D Euler Riemann#

This problem solves the 2D conservative Euler equations

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

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

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

  • Various initial conditions are supported:

    • icId==1:

      • See page 15 of paper1

        \(\left\{\begin{matrix}\rho = 0.5313, u = 0, v = 0, p = 0.4; & x\geq 1/2, y\geq 1/2\\ \rho = 1, u = 0.7276, v = 0, p = 1; & x<1/2, y\geq 1/2 \\ \rho = 4/5, u = 0, v = 0, p = 1; & x<1/2, y<1/2 \\ \rho = 1, u = 0, v = 0.7276, p = 1;& x>1/2, y<1/2 \end{matrix}\right.\)

      • This IC is used to create the corresponding initial conditions in conservative variables.

      • Time integration is performed for \(t \in (0, 1/5)\)

    • icId==2:

      • See configuration 3 of paper2

        \(\left\{\begin{matrix}\rho = 1.5, u = 0, v = 0, p = 1.5; & x\geq 4/5, y\geq 4/5\\ \rho = 0.5323, u = 1.206, v = 0, p = 0.3; & x<4/5, y\geq 4/5 \\ \rho = 0.138, u = 1.206, v = 1.206, p = 0.029; &x<4/5, y<4/5 \\ \rho = 0.5323, u = 0, v = 1.206, p = 0.3;& x>4/5, y<4/5 \end{matrix}\right.\)

      • This IC is used to create the corresponding initial conditions in conservative variables.

      • Time integration is performed for \(t \in (0, 4/5)\)

  • By default, \(\gamma = 1.4\)

  • Domain is \([0, 1]^2\) with homogeneous Neumann on all boundaries

Mesh#

python3 pressio-demoapps/meshing_scripts/create_full_mesh_for.py \
        --problem riemann2d_s<stencilSize> -n Nx Ny --outDir <destination-path>

where

  • Nx, Ny is the number of cells you want along \(x\) and \(y\) respectively

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

  • <destination-path> is 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/euler2d.hpp"

int main(){
  namespace pda     = pressiodemoapps;

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

  const auto probId = pda::Euler2d::Riemann;
  const auto scheme = pda::InviscidFluxReconstruction::FirstOrder; //or Weno3, Weno5
  auto problem      = pda::create_problem_eigen(meshObj, probId, scheme [, icId]);
  auto state      = problem.initialCondition();
}

Where the icId is an integer identifying the initial condition above.

Python synopsis#

import pressiodemoapps as pda

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

probId  = pda.Euler2d.Riemann
scheme  = pda.InviscidFluxReconstruction.FirstOrder # or Weno3, Weno5
problem = pda.create_problem(meshObj, probId, scheme [, icId])
state   = problem.initialCondition()

Where the icId is an integer identifying the initial condition above.

Sample Plot#

Representative density plot at \(t=4/5\) using icId=2 initial conditions with Weno5, SSPRK3 time integration:

Alternative text