Euler Physics

Describe the equation being solved here

\[\frac{\partial q}{\partial t} = - \nabla \cdot F(q) + S\]

Where $q$ are the conservative variables and $F$ is the Euler flux.

\[q = \begin{bmatrix} \rho, \rho u, \rho v, \rho w, e \end{bmatrix}\]

where $\rho$ is density, $u$ is the x velocity, $v$ is the y velocity, $w$ is the z velocity and $e$ is the energy.

The calloricaly perfect ideal gas law is used to close the system (TODO ref calcPressure)

The x-y-z components of the Euler flux are:

\[\begin{bmatrix} \rho u & \rho v & \rho w \\ \rho u^{2} + p & \rho u v & \rho u w \\ \rho u v & \rho v^2 + p & \rho v w \\ \rho u w & \rho v w & \rho w^2 + p \\ (e + p)u & (e + p)v & (e + p)w \end{bmatrix}\]

TODO: describe the physical quantities (include units)

Discretizations

The code currently implements three kinds of discretizations.

The first is a standard discontinuous-Galerkin scheme using SBP operators and uses the numerical flux functions in flux functions section.

The second scheme is an entropy stable discretization that uses a Hadamard product between the SBP operator matrices and matrices of flux values. The face integrals for the entropy stable scheme are described on the face element integrals page.

The third scheme is a staggerd grid approach based on the entropy stable scheme. It uses all the same mechanics as the entropy stable scheme, but requires interpolating data back and forth between the solution and flux grids. The functions for doing the interpolation are listed on the relevent pages for the volume and face integrals.