Stabilized methods for high-speed compressible flows: toward hypersonic simulations

Abstract

A stabilized finite element framework for high-speed compressible flows is presented. The Streamline-Upwind/Petrov–Galerkin formulation augmented with discontinuity-capturing (DC) are the main constituents of the framework that enable accurate, efficient, and stable simulations in this flow regime. Full- and reduced-energy formulations are employed for this class of flow problems and their relative accuracy is assessed. In addition, a recently developed DC formulation is presented and is shown to be particularly well suited for hypersonic flows. Several verification and validation cases, ranging from 1D to 3D flows and supersonic to the hypersonic regimes, show the excellent performance of the proposed framework and set the stage for its deployment on more advanced applications.

Appendix A

The matrices used for Navier–Stokes equations of compressible flows with full energy equation are given by

$$\begin{aligned} \tilde{{\mathbf {A}}}_0 = \begin{bmatrix} \rho \beta _{T} &{}0&{}0 &{}0 &{} -\rho \alpha _{p}\\ \rho \beta _{T} u_1 &{}\rho &{}0 &{}0 &{} -\rho \alpha _{p}u_1\\ \rho \beta _{T} u_2 &{}0&{}\rho &{}0 &{} -\rho \alpha _{p}u_2\\ \rho \beta _{T} u_3 &{}0&{}0 &{}\rho &{} -\rho \alpha _{p}u_3\\ \rho \beta _{T} e_\text {tot} &{}\rho u_1&{}\rho u_2 &{}\rho u_3 &{} \rho \left( -\alpha _{p} e_\text {tot} + c_\text {v} \right) \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.1)

where $\beta _{T} = 1/p$, $\alpha _{p} = 1/T$.

Its inverse $\tilde{{\mathbf {A}}}_0^{-1} = {\mathbf {Y}}_{,\tilde{{\mathbf {U}}}}$ is given by

$$\begin{aligned} \tilde{{\mathbf {A}}}_0^{-1} = \begin{bmatrix} \dfrac{-\alpha _{p}e_\text {tot} +\alpha _{p} \Vert {\mathbf {u}}\Vert ^2+ c_\text {v}}{\rho \beta _{T} c_\text {v}}&{}-\dfrac{\alpha _pu_1}{\rho \beta _{T} c_\text {v}}&{}-\dfrac{\alpha _pu_2}{\rho \beta _{T} c_\text {v}} &{}-\dfrac{\alpha _pu_3}{\rho \beta _{T} c_\text {v}} &{} \dfrac{\alpha _p}{\rho \beta _{T} c_\text {v}}\\ -\dfrac{u_1}{\rho } &{}\dfrac{1}{\rho }&{}0 &{}0 &{} 0\\ -\dfrac{u_2}{\rho } &{}0&{}\dfrac{1}{\rho } &{}0 &{} 0\\ -\dfrac{u_3}{\rho } &{}0&{}0 &{}\dfrac{1}{\rho }&{} 0\\ \dfrac{ \Vert {\mathbf {u}}\Vert ^2 - e_\text {tot}}{\rho c_\text {v}}&{}-\dfrac{u_1}{\rho c_\text {v}}&{}-\dfrac{u_2}{\rho c_\text {v}}&{}-\dfrac{u_3}{\rho c_\text {v}}&{} \dfrac{1}{\rho c_\text {v}}\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.2)

We then give the details of the Euler Jacobian matrices by

$$\begin{aligned} \tilde{{\mathbf {A}}}_1^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_1 &{} \rho &{}0 &{}0 &{} -\rho \alpha _{p} u_1\\ \rho \beta _{T} u_1^2 &{}2\rho u_1&{}0 &{}0 &{} -\rho \alpha _{p}u_1^2\\ \rho \beta _{T} u_1 u_2 &{}\rho u_2&{}\rho u_1&{}0 &{} -\rho \alpha _{p} u_1 u_2\\ \rho \beta _{T} u_1 u_3 &{}\rho u_3&{}0 &{}\rho u_1&{} -\rho \alpha _{p} u_1 u_3\\ \left( \rho \beta _{T} e_\text {tot} + 1\right) u_1 &{}\rho \left( e_\text {tot} + u_1^2 \right) + p&{}\rho u_1 u_2 &{}\rho u_1 u_3 &{} \rho \left( -\alpha _{p} e_\text {tot} + c_\text {v} \right) u_1\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.3)

$$\begin{aligned} \tilde{{\mathbf {A}}}_2^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_2 &{}0&{} \rho &{}0 &{} -\rho \alpha _{p} u_2\\ \rho \beta _{T} u_1 u_2 &{}\rho u_2&{}\rho u_1 &{}0 &{} -\rho \alpha _{p}u_1 u_2\\ \rho \beta _{T} u_2^2 &{} 0 &{}2\rho u_2&{}0 &{} -\rho \alpha _{p} u_2^2\\ \rho \beta _{T} u_2 u_3 &{}0 &{}\rho u_3&{}\rho u_2&{} -\rho \alpha _{p} u_2 u_3\\ \left( \rho \beta _{T} e_\text {tot} + 1\right) u_2 &{}\rho u_1 u_2 &{}\rho \left( e_\text {tot} + u_2^2 \right) + p&{}\rho u_2 u_3 &{} \rho \left( -\alpha _{p} e_\text {tot} + c_\text {v} \right) u_2\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.4)

$$\begin{aligned} \tilde{{\mathbf {A}}}_3^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_3 &{} 0 &{}0 &{} \rho &{} -\rho \alpha _{p} u_3\\ \rho \beta _{T} u_1 u_3 &{}\rho u_3&{}0 &{}\rho u_1&{} -\rho \alpha _{p}u_1 u_3\\ \rho \beta _{T} u_2 u_3 &{} 0 &{}\rho u_3&{}\rho u_2 &{} -\rho \alpha _{p} u_2 u_3\\ \rho \beta _{T} u_3^2 &{}0 &{}0&{}2\rho u_3&{} -\rho \alpha _{p} u_3^2\\ \left( \rho \beta _{T} e_\text {tot} + 1\right) u_3 &{}\rho u_1 u_3 &{}\rho u_2 u_3 &{}\rho \left( e_\text {tot} + u_3^2 \right) + p&{} \rho \left( -\alpha _{p} e_\text {tot} + c_\text {v} \right) u_3\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.5)

$$\begin{aligned} \tilde{{\mathbf {A}}}_1^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.6)

$$\begin{aligned} \tilde{{\mathbf {A}}}_2^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.7)

$$\begin{aligned} \tilde{{\mathbf {A}}}_3^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.8)

Note that $\tilde{{\mathbf {A}}}_i = \tilde{{\mathbf {A}}}_i^{\text {adv} \backslash p} + \tilde{{\mathbf {A}}}_i^{p}$.

Finally, we give the diffusive matrices by

$$\begin{aligned} \tilde{{\mathbf {K}}}_{11}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 2\mu +\lambda &{} 0&{} 0&{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{} \left( 2\mu +\lambda \right) u_1&{} \mu u_2&{} \mu u_3&{} \kappa \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.9)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{12}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} \lambda &{} 0&{} 0\\ 0 &{} \mu &{}0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \mu u_2&{} \lambda u_1&{} 0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.10)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{13}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} \lambda &{} 0\\ 0 &{} 0&{}0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{} \mu u_3&{}0&{} \lambda u_1 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.11)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{21}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} \lambda &{}0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \lambda u_2&{} \mu u_1&{}0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.12)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{22}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{} 0&{} 2\mu +\lambda &{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{} \mu u_1&{} \left( 2\mu +\lambda \right) u_2 &{} \mu u_3&{} \kappa \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.13)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{23}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{}0&{} \lambda &{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} 0&{} \mu u_3&{}\lambda u_2 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.14)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{31}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{} 0 &{}0&{} 0&{} 0\\ 0 &{} \lambda &{} 0&{} 0&{} 0\\ 0 &{} \lambda u_3&{} 0&{} \mu u_1 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.15)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{32}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{}0&{} \mu &{} 0\\ 0 &{} 0&{} \lambda &{} 0&{} 0\\ 0 &{} 0&{} \lambda u_3&{}\mu u_2 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.16)

$$\begin{aligned} \tilde{{\mathbf {K}}}_{33}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{} 0&{}\mu &{} 0&{} 0\\ 0 &{} 0&{} 0&{} 2\mu +\lambda &{} 0\\ 0 &{} \mu u_1&{} \mu u_2&{} \left( 2\mu +\lambda \right) u_3 &{} \kappa \\ \end{bmatrix}\text {.} \end{aligned}$$

(A.17)

For the Navier–Stokes equations with reduced energy formulation, the matrices corresponding to pressure-primitive variables are as follows:

The matrix ${\mathbf {A}}_0 = {\mathbf {U}}_{,{\mathbf {Y}}}$ is given by

$$\begin{aligned} {\mathbf {A}}_0 = \begin{bmatrix} \rho \beta _{T} &{}0&{}0 &{}0 &{} -\rho \alpha _{p}\\ \rho \beta _{T} u_1 &{}\rho &{}0 &{}0 &{} -\rho \alpha _{p}u_1\\ \rho \beta _{T} u_2 &{}0&{}\rho &{}0 &{} -\rho \alpha _{p}u_2\\ \rho \beta _{T} u_3 &{}0&{}0 &{}\rho &{} -\rho \alpha _{p}u_3\\ \rho \beta _{T} e &{}0&{}0&{}0&{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.18)

It’s inverse ${\mathbf {A}}_0^{-1} = {\mathbf {Y}}_{,{\mathbf {U}}}$ is given by

$$\begin{aligned} {\mathbf {A}}_0^{-1} = \begin{bmatrix} 0&{}0&{}0 &{}0&{} \dfrac{\alpha _p}{\rho \beta _{T} c_\text {v}}\\ -\dfrac{u_1}{\rho } &{}\dfrac{1}{\rho }&{}0 &{}0 &{} 0\\ -\dfrac{u_2}{\rho } &{}0&{}\dfrac{1}{\rho } &{}0 &{} 0\\ -\dfrac{u_3}{\rho } &{}0&{}0 &{}\dfrac{1}{\rho }&{} 0\\ -\dfrac{T}{\rho }&{}0&{}0&{}0&{} \dfrac{1}{\rho c_\text {v}}\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.19)

The Euler–Jacobian matrices are given by

$$\begin{aligned} {\mathbf {A}}_1^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_1 &{} \rho &{}0 &{}0 &{} -\rho \alpha _{p} u_1\\ \rho \beta _{T} u_1^2 &{}2\rho u_1&{}0 &{}0 &{} -\rho \alpha _{p}u_1^2\\ \rho \beta _{T} u_1 u_2 &{}\rho u_2&{}\rho u_1&{}0 &{} -\rho \alpha _{p} u_1 u_2\\ \rho \beta _{T} u_1 u_3 &{}\rho u_3&{}0 &{}\rho u_1&{} -\rho \alpha _{p} u_1 u_3\\ \rho \beta _{T} e u_1 &{}\rho e &{}0&{}0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.20)

$$\begin{aligned} {\mathbf {A}}_2^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_2 &{}0&{} \rho &{}0 &{} -\rho \alpha _{p} u_2\\ \rho \beta _{T} u_1 u_2 &{}\rho u_2&{}\rho u_1 &{}0 &{} -\rho \alpha _{p}u_1 u_2\\ \rho \beta _{T} u_2^2 &{} 0 &{}2\rho u_2&{}0 &{} -\rho \alpha _{p} u_2^2\\ \rho \beta _{T} u_2 u_3 &{}0 &{}\rho u_3&{}\rho u_2&{} -\rho \alpha _{p} u_2 u_3\\ \rho \beta _{T} e u_2 &{}0 &{}\rho e&{}0&{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.21)

$$\begin{aligned} {\mathbf {A}}_3^{\text {adv} \backslash p}&= \begin{bmatrix} \rho \beta _{T} u_3 &{} 0 &{}0 &{} \rho &{} -\rho \alpha _{p} u_3\\ \rho \beta _{T} u_1 u_3 &{}\rho u_3&{}0 &{}\rho u_1&{} -\rho \alpha _{p}u_1 u_3\\ \rho \beta _{T} u_2 u_3 &{} 0 &{}\rho u_3&{}\rho u_2 &{} -\rho \alpha _{p} u_2 u_3\\ \rho \beta _{T} u_3^2 &{}0 &{}0&{}2\rho u_3&{} -\rho \alpha _{p} u_3^2\\ \rho \beta _{T} e u_3 &{} 0 &{} 0 &{} \rho e &{} 0 \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.22)

$$\begin{aligned} {\mathbf {A}}_1^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.23)

$$\begin{aligned} {\mathbf {A}}_2^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.24)

$$\begin{aligned} {\mathbf {A}}_3^{p}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 1 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.25)

$$\begin{aligned} {\mathbf {A}}_1^{\text {sp}}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} p-\tau _{11}&{} -\tau _{12}&{} -\tau _{13}&{} 0\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.26)

$$\begin{aligned} {\mathbf {A}}_2^{\text {sp}}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} -\tau _{21}&{}p -\tau _{22}&{} -\tau _{23}&{} 0\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.27)

$$\begin{aligned} {\mathbf {A}}_3^{\text {sp}}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} -\tau _{31}&{} -\tau _{32}&{}p -\tau _{33}&{} 0\\ \end{bmatrix}\text {.} \end{aligned}$$

(A.28)

The diffusivity matrices are given by

$$\begin{aligned} {\mathbf {K}}_{11}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 2\mu +\lambda &{} 0&{} 0&{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{}0 &{} 0&{} 0&{} \kappa \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.29)

$$\begin{aligned} {\mathbf {K}}_{12}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} \lambda &{} 0&{} 0\\ 0 &{} \mu &{}0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{}0&{}0&{} 0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.30)

$$\begin{aligned} {\mathbf {K}}_{13}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} \lambda &{} 0\\ 0 &{} 0&{}0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{}0&{}0&{}0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.31)

$$\begin{aligned} {\mathbf {K}}_{21}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} \lambda &{}0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{}0&{}0&{}0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.32)

$$\begin{aligned} {\mathbf {K}}_{22}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{} 0&{} 2\mu +\lambda &{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{}0&{}0&{}0&{} \kappa \\ \end{bmatrix}\text {,} \end{aligned}$$

(A.33)

$$\begin{aligned} {\mathbf {K}}_{23}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{}0&{} \lambda &{} 0\\ 0 &{} 0&{} \mu &{} 0&{} 0\\ 0 &{} 0&{} 0&{}0 &{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.34)

$$\begin{aligned} {\mathbf {K}}_{31}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} \mu &{} 0\\ 0 &{} 0 &{}0&{} 0&{} 0\\ 0 &{} \lambda &{} 0&{} 0&{} 0\\ 0 &{} 0&{}0&{}0&{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.35)

$$\begin{aligned} {\mathbf {K}}_{32}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{}0&{} \mu &{} 0\\ 0 &{} 0&{} \lambda &{} 0&{} 0\\ 0 &{} 0&{}0&{}0&{}0\\ \end{bmatrix}\text {,} \end{aligned}$$

(A.36)

$$\begin{aligned} {\mathbf {K}}_{33}&= \begin{bmatrix} 0 &{} 0&{} 0&{} 0&{} 0\\ 0 &{} \mu &{} 0&{} 0&{} 0\\ 0 &{} 0&{}\mu &{} 0&{} 0\\ 0 &{} 0&{} 0&{} 2\mu +\lambda &{} 0\\ 0 &{} 0&{}0&{}0 &{} \kappa \\ \end{bmatrix}\text {.} \end{aligned}$$

(A.37)

The matrices for the conservation variables may be obtained from the corresponding matrices for the pressure-primitive variables using the following transformations: $\hat{{\mathbf {A}}}_i={\mathbf {A}}_i{\mathbf {A}}_0^{-1}$, $\hat{{\mathbf {A}}}_i^{\text {adv} \backslash p}={\mathbf {A}}_i^{\text {adv} \backslash p}{\mathbf {A}}_0^{-1}$, $\hat{{\mathbf {A}}}_i^{p}={\mathbf {A}}_i^{p}{\mathbf {A}}_0^{-1}$, $\hat{{\mathbf {A}}}_i^{\text {sp}}={\mathbf {A}}_i^{\text {sp}}{\mathbf {A}}_0^{-1}$, and $\hat{{\mathbf {K}}}_{ij}={\mathbf {K}}_{ij}{\mathbf {A}}_0^{-1}$