Entropy Stable Finite Difference Schemes for Chew, Goldberger and Low Anisotropic Plasma Flow Equations

Abstract

In this article, we consider the Chew, Goldberger and Low (CGL) plasma flow equations, which is a set of nonlinear, non-conservative hyperbolic PDEs modeling anisotropic plasma flows. These equations incorporate the double adiabatic approximation for the evolution of the pressure, making them very valuable for plasma physics, space physics, and astrophysical applications. We first present the entropy analysis for the weak solutions. We then propose entropy-stable finite-difference schemes for the CGL equations. The key idea is to reformulate the CGL equations by rewriting some of the conservative terms in the non-conservation form. The conservative part of the reformulated equations is very similar to the magnetohydrodynamics (MHD) equations which is then symmetrized using Godunov’s symmetrization process for the MHD equations. The resulting equations are in the form where the conservative part combined with non-conservative Godunov’s terms is compatible with the entropy equation and the rest of the non-conservative terms do not contribute to the entropy equations. The final set of reformulated equations is then discretized by designing entropy conservative numerical flux and entropy diffusion operator based on the entropy scaled eigenvectors of the conservative part. We then prove the semi-discrete entropy stability of the schemes for the reformulated CGL equations. The schemes are then tested using several test problems derived from the corresponding MHD test cases.

Appendices

Entropy Scaled Right Eigenvectors for Conservative Part of the Reformulated CGL System

In this section, we present expressions for the right eigenvectors and scaled right eigenvectors for the conservative system

$$\begin{aligned} \frac{\partial {{\textbf {U}}}}{\partial t} + {\mathcal {A}}_x({{\textbf {U}}})\frac{\partial {{\textbf {U}}}}{\partial x} = 0,~~~~~~~~\text {where,}~ {\mathcal {A}}_x({{\textbf {U}}})=\frac{\partial {{\textbf {f}}}_x}{\partial {{\textbf {U}}}} + \phi '({{\textbf {V}}})^\top B_x'({{\textbf {U}}}) \end{aligned}$$

(60)

The right eigenvectors are described in Section A.1, while the expressions of scaled eigenvectors have been demonstrated in Section A.2. We will do all the analysis in the term of primitive variable ${{\textbf {W}}} = \{\rho ,u_x,u_y,u_z,p_\parallel ,p_{\perp },B_x,B_y,B_z\}$.

1.1 Eigenvalues and Right Eigenvectors

To derive the eigenvalues and right eigenvectors, it is useful to transform the system (60) in terms of the primitive variables ${{\textbf {W}}}$. The set of eigenvalues $\Lambda _1$ of the jacobian matrix ${\mathcal {A}}_x$ are given as

$$\begin{aligned}&\lambda _{1,2,3,4,5,6,7,8}=u_x,~ u_x, ~u_x\pm v_{ax},~u_x\pm c_f,~u_x\pm c_s, \end{aligned}$$

where,

$$\begin{aligned}&v^2_{ax}= \frac{B_x^2}{\rho },~v^2_a=\frac{B^2}{\rho }, ~a^2=\frac{2P_\perp }{\rho },&\\ &c^2_{f,s}=\frac{1}{2}\bigg [(v_a^2+a^2)\pm \sqrt{(v_a^2+a^2)^2-4v^2_{ax}a^2}\bigg ]. \end{aligned}$$

The corresponding right eigenvectors are

$$\begin{aligned}R_{\lambda _1}= & \begin{pmatrix} 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{pmatrix},R_{\lambda _2}= \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0 \end{pmatrix},R_{\lambda _{3,4}}= \begin{pmatrix} 0\\ 0\\ \pm {\beta _z}\\ \mp {\beta _y}\\ 0\\ 0\\ -{\beta _z}sgn(B_x)\sqrt{\rho }\\ {\beta _y}sgn(B_x)\sqrt{\rho }\\ \end{pmatrix},\\ R_{\lambda _{5,6}}= & \begin{pmatrix} \alpha _f \rho \\ \pm \alpha _f c_f\\ \mp \alpha _s c_s \beta _y sgn(B_x)\\ \mp \alpha _s c_s \beta _z sgn(B_x)\\ \alpha _f p_\parallel \\ \alpha _f \rho a^2\\ \alpha _s a\beta _y\sqrt{\rho }\\ \alpha _s a\beta _z\sqrt{\rho } \end{pmatrix},R_{\lambda _{7,8}}= \begin{pmatrix} \alpha _s \rho \\ \pm \alpha _s c_s\\ \pm \alpha _f c_f \beta _y sgn(B_x)\\ \pm \alpha _fc_f \beta _z sgn(B_x)\\ \alpha _s p_\parallel \\ \alpha _s \rho a^2\\ -\alpha _f a\beta _y\sqrt{\rho } \\ -\alpha _f a\beta _z\sqrt{\rho } \end{pmatrix}, \end{aligned}$$

where,

$$\begin{aligned} \alpha _f^2=\frac{a^2-c_s^2}{c_f^2-c_s^2},~\alpha _s^2=\frac{c_f^2-a^2}{c_f^2-c_s^2},~ \beta _y=\frac{B_y}{\sqrt{B_y^2+B_z^2}},~\beta _z=\frac{B_z}{\sqrt{B_y^2+B_z^2}}. \end{aligned}$$

Above all, eigenvectors are linearly independent. The above-defined right eigenvectors are singular in a variety of cases as listed by Balsara and Roe in [3]. Similarly, in y-direction, the set of eigenvalues $\Lambda _2$ of the jacobian matrix $\frac{\partial {{\textbf {f}}}_{y}}{\partial {{\textbf {U}}}}$ are given as

$$\begin{aligned}&\lambda _{1,2,3,4,5,6,7,8}=u_y,~ u_y, ~u_y\pm v_{ay},~u_y\pm c_f,~u_x\pm c_s, \end{aligned}$$

where,

$$\begin{aligned}&v^2_{ay}= \frac{B_y^2}{\rho },~v^2_a=\frac{B^2}{\rho }, ~a^2=\frac{2P_\perp }{\rho },&\\ &c^2_{f,s}=\frac{1}{2}\bigg [(v_a^2+a^2)\pm \sqrt{(v_a^2+a^2)^2-4v^2_{ay}a^2}\bigg ]. \end{aligned}$$

the corresponding right eigenvectors are

$$\begin{aligned}R_{\lambda _1}= & \begin{pmatrix} 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{pmatrix},R_{\lambda _2}= \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0 \end{pmatrix},R_{\lambda _{3,4}}= \begin{pmatrix} 0\\ \pm {\beta _z}\\ 0\\ \mp {\beta _x}\\ 0\\ 0\\ -{\beta _z}sgn(B_y)\sqrt{\rho }\\ {\beta _x}sgn(B_y)\sqrt{\rho }\\ \end{pmatrix},\\ R_{\lambda _{5,6}}= & \begin{pmatrix} \alpha _f \rho \\ \mp \alpha _s c_s \beta _x sgn(B_y)\\ \pm \alpha _f c_f\\ \mp \alpha _s c_s \beta _z sgn(B_y)\\ \alpha _f p_\parallel \\ \alpha _f \rho a^2\\ \alpha _s a\beta _x\sqrt{\rho }\\ \alpha _s a\beta _z\sqrt{\rho } \end{pmatrix},R_{\lambda _{7,8}}= \begin{pmatrix} \alpha _s \rho \\ \pm \alpha _f c_f \beta _x sgn(B_y)\\ \pm \alpha _s c_s\\ \pm \alpha _fc_f \beta _z sgn(B_y)\\ \alpha _s p_\parallel \\ \alpha _s \rho a^2\\ -\alpha _f a\beta _x\sqrt{\rho } \\ -\alpha _f a\beta _z\sqrt{\rho } \end{pmatrix}, \end{aligned}$$

where,

$$\begin{aligned} \alpha _f^2=\frac{a^2-c_s^2}{c_f^2-c_s^2},~\alpha _s^2=\frac{c_f^2-a^2}{c_f^2-c_s^2},~ \beta _x=\frac{B_x}{\sqrt{B_x^2+B_z^2}},~\beta _z=\frac{B_z}{\sqrt{B_x^2+B_z^2}}. \end{aligned}$$

1.2 Entropy Scaled Right Eigenvectors

In this section, we have calculated the entropy scaled right eigenvectors for the case of x-direction. Consider the conservative part of the CGL system (60). The eigenvalues and Right eigenvectors in terms of primitive of the Jacobian matrix ${\mathcal {A}}_1$ are described in (A.1). The right eigenvector matrix $R^x$ for the matrix ${\mathcal {A}}_1$ in terms of conservative variable is given by the relation

$$\begin{aligned} R^x = \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {W}}}} R_{{{\textbf {W}}}}^{x} \end{aligned}$$

where, $\frac{\partial {{\textbf {U}}}}{\partial {{\textbf {W}}}}$ is the Jacobian matrix for the change of variable, given by

$$\begin{aligned} \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {W}}}}= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ u_x & \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ u_y & 0 & \rho & 0 & 0 & 0 & 0 & 0 & 0\\ u_z & 0 & 0 & \rho & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \frac{{{\textbf {u}}}^2}{2} & \rho u_x & \rho u_y & \rho u_z & \frac{1}{2} & 1 & B_x & B_y & B_z\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. \end{aligned}$$

and the matrix $R_{{{\textbf {W}}}}^{x}$ is given by,

$$\begin{aligned} \begin{pmatrix} \alpha _f \rho & \alpha _s \rho & 0 & 1 & 0 & 0 & 0 & \alpha _s \rho & \alpha _f \rho \\ -\alpha _f c_f & -\alpha _s c_s & 0 & 0 & 0 & 0 & 0 & \alpha _s c_s & \alpha _f c_f \\ \alpha _s \beta _y c_s {\mathcal {S}}_{x} & -\alpha _f \beta _y c_f {\mathcal {S}}_{x} & -\beta _z & 0 & 0 & 0 & \beta _z & \alpha _f \beta _y c_f {\mathcal {S}}_{x}& -\alpha _s\beta _y c_s {\mathcal {S}}_{x}\\ \alpha _s \beta _z c_s {\mathcal {S}}_{x} & -\alpha _f \beta _z c_f {\mathcal {S}}_{x} & \beta _y & 0 & 0 & 0 & -\beta _y & \alpha _f \beta _z c_f {\mathcal {S}}_{x}& -\alpha _s \beta _z c_s {\mathcal {S}}_{x} \\ \alpha _f p_\parallel & \alpha _s p_\parallel & 0 & 0 & 0 & 1 & 0 & \alpha _s p_\parallel & \alpha _f p_\parallel \\ a^2 \alpha _f \rho & a^2 \alpha _s \rho & 0 & 0 & 0 & 0 & 0 & a^2 \alpha _s \rho & a^2 \alpha _f \rho \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ a \alpha _s \beta _y \sqrt{\rho } & -a \alpha _f \beta _y \sqrt{\rho } & -\beta _z \sqrt{\rho } {\mathcal {S}}_{x} & 0 & 0 & 0 & -\beta _z \sqrt{\rho } {\mathcal {S}}_{x} & -a \alpha _f \beta _y \sqrt{\rho } & a \alpha _s \beta _y \sqrt{\rho } \\ a \alpha _s \beta _z \sqrt{\rho } & -a \alpha _f \beta _z \sqrt{\rho } & \beta _y \sqrt{\rho } {\mathcal {S}}_{x} & 0 & 0 & 0 & \beta _y \sqrt{\rho } {\mathcal {S}}_{x} & -a \alpha _f \beta _z \sqrt{\rho } & a \alpha _s \beta _z \sqrt{\rho } \\ \end{pmatrix}. \end{aligned}$$

where, ${\mathcal {S}}_{x} = sgn(B_x)$. We need to find a scaling matrix $T^x$ such that the scaled right eigenvector matrix $\tilde{R}^x=R^x T^x$ satisfies

$$\begin{aligned} \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {V}}}} = \tilde{R}^x \tilde{R}^{x\top } \end{aligned}$$

(61)

where ${{\textbf {V}}}$ is the entropy variable vector as in Eqn. (12). We follow the Barth scaling process [6] to scale the right eigenvectors. The scaling matrix $T^x$ is the square root of $Y^x$, where $Y^x$ has the expression

$$\begin{aligned} Y^x = (R_{{{\textbf {W}}}}^{x})^{-1} \frac{\partial {{\textbf {W}}}}{\partial {{\textbf {V}}}}\left( \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {W}}}}\right) ^{-\top }(R_{{{\textbf {W}}}}^{x}) ^{-\top } \end{aligned}$$

which results in

$$\begin{aligned} Y^x = \begin{pmatrix} \frac{1}{8\rho } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \frac{1}{8\rho } & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \frac{p_{\perp }}{4\rho ^2} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{\rho }{4} & 0 & \frac{p_\parallel }{4} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{p_{\perp }}{2\rho } & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{p_\parallel }{4} & 0& \frac{5 p_\parallel ^2}{4\rho } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{p_{\perp }}{4\rho ^2}& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{8\rho } & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{8\rho } \end{pmatrix}. \end{aligned}$$

Then matrix $T^x$ is,

$$\begin{aligned} \begin{pmatrix} \frac{1}{2\sqrt{2\rho }} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \frac{1}{2\sqrt{2\rho }} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \frac{\sqrt{p_{\perp }}}{2\rho } & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{\sqrt{\rho }(2 p_\parallel + \rho )}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{\frac{p_{\perp }}{2\rho }} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & \frac{p_\parallel (5 p_\parallel + 2 \rho )}{2 \sqrt{\rho (5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2)}}& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{p_{\perp }}}{2\rho }& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2\sqrt{2\rho }} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2\sqrt{2\rho }} \end{pmatrix}. \end{aligned}$$

Remark 3

For finding the square root of $3\times 3$ matrix

$$\begin{aligned} M = \begin{pmatrix} a & 0 & b\\ 0 & e & 0 \\ c & 0 & d \end{pmatrix} \end{aligned}$$

where a , b , c, d and e are real or complex numbers. we apply the formula,

$$\begin{aligned} R = \begin{pmatrix} \frac{a+s}{t} & 0 & \frac{b}{t}\\ 0 & \sqrt{e} & 0\\ \frac{c}{t} & 0 & \frac{d+s}{t} \end{pmatrix}. \end{aligned}$$

where, $\delta = ad-bc$, $\tau = a+d$ and $s=\sqrt{\delta }$, $t=\sqrt{\tau +2s}$. We apply this formula if $t \ne 0$.

Remark 4

We proceed similarly in the y-direction, the eigenvalues and Right eigenvectors in terms of primitive of the Jacobian matrix $\frac{\partial {{\textbf {f}}}_{2}}{\partial {{\textbf {U}}}}$ describe in (A.1). The right eigenvector matrix is given by the relation

$$\begin{aligned} R^y = \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {W}}}} R_{{{\textbf {W}}}}^{y} \end{aligned}$$

where, $R_{{{\textbf {W}}}}^{y}$ is given below

$$\begin{aligned} \begin{pmatrix} \alpha _f \rho & \alpha _s \rho & 0 & 1 & 0 & 0 & 0 & \alpha _s \rho & \alpha _f \rho \\ \alpha _s \beta _x c_s {\mathcal {S}}_{y} & -\alpha _f \beta _x c_f {\mathcal {S}}_{y} & -\beta _z & 0 & 0 & 0 & \beta _z & \alpha _f \beta _x c_f {\mathcal {S}}_{y}& -\alpha _s\beta _x c_s {\mathcal {S}}_{y}\\ -\alpha _f c_f & -\alpha _s c_s & 0 & 0 & 0 & 0 & 0 & \alpha _s c_s & \alpha _f c_f \\ \alpha _s \beta _z c_s {\mathcal {S}}_{y} & -\alpha _f \beta _z c_f {\mathcal {S}}_{y} & \beta _x & 0 & 0 & 0 & -\beta _x & \alpha _f \beta _z c_f {\mathcal {S}}_{y}& -\alpha _s \beta _z c_s {\mathcal {S}}_{y}\\ \alpha _f p_\parallel & \alpha _s p_\parallel & 0 & 0 & 0 & 1 & 0 & \alpha _s p_\parallel & \alpha _f p_\parallel \\ a^2 \alpha _f \rho & a^2 \alpha _s \rho & 0 & 0 & 0 & 0 & 0 & a^2 \alpha _s \rho & a^2 \alpha _f \rho \\ a \alpha _s \beta _x \sqrt{\rho } & -a \alpha _f \beta _x \sqrt{\rho } & -\beta _z \sqrt{\rho } {\mathcal {S}}_{y} & 0 & 0 & 0 & -\beta _z \sqrt{\rho } {\mathcal {S}}_{y} & -a \alpha _f \beta _x \sqrt{\rho } & a \alpha _s \beta _x \sqrt{\rho } \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ a \alpha _s \beta _z \sqrt{\rho } & -a \alpha _f \beta _z \sqrt{\rho } & \beta _x \sqrt{\rho } {\mathcal {S}}_{y} & 0 & 0 & 0 & \beta _x \sqrt{\rho } {\mathcal {S}}_{y} & -a \alpha _f \beta _z \sqrt{\rho } & a \alpha _s \beta _z \sqrt{\rho } \\ \end{pmatrix} \end{aligned}$$

where, ${\mathcal {S}}_{y} = sgn(B_y)$. Accordingly, we obtain the scaling matrix $T^y$ as,

$$\begin{aligned} \begin{pmatrix} \frac{1}{2\sqrt{2\rho }} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \frac{1}{2\sqrt{2\rho }} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \frac{\sqrt{p_{\perp }}}{2\rho } & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{\sqrt{\rho }(2 p_\parallel + \rho )}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{\frac{p_{\perp }}{2\rho }} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}} & 0 & \frac{p_\parallel (5 p_\parallel + 2 \rho )}{2 \sqrt{\rho (5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2)}}& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{p_{\perp }}}{2\rho }& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2\sqrt{2\rho }} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2\sqrt{2\rho }} \end{pmatrix}. \end{aligned}$$

1.3 Entropy Scaled Right Eigenvectors in x and y Direction

In x direction, the entropy-scaled right eigenvectors corresponding to the above eigenvalues in the term of primitive variables are the following,

$$\begin{aligned}{\tilde{R}}_{\lambda _1}= & \begin{pmatrix} \frac{\sqrt{\rho }(2 p_\parallel + \rho )}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ 0 \end{pmatrix},{\tilde{R}}_{\lambda _2}= \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \sqrt{\frac{p_{\perp }}{2 \rho }}\\ 0\\ 0 \end{pmatrix},{\tilde{R}}_{\lambda _3}= \begin{pmatrix} \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ \frac{p_\parallel (5 p_\parallel + 2 \rho )}{2 \sqrt{\rho (5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2)}}\\ 0\\ 0\\ 0\\ 0 \end{pmatrix},\\ {\tilde{R}}_{\lambda _{4,5}}= & \frac{1}{2}\begin{pmatrix} 0\\ 0\\ \pm \frac{\sqrt{p_{\perp }}}{\rho }{\beta _z}\\ \mp \frac{\sqrt{p_{\perp }}}{\rho }{\beta _y}\\ 0\\ 0\\ 0\\ -\sqrt{\frac{p_{\perp }}{\rho }}{\beta _z}\\ \sqrt{\frac{p_{\perp }}{\rho }}{\beta _y}\\ \end{pmatrix},{\tilde{R}}_{\lambda _{6,7}}= \frac{1}{2\sqrt{2}}\begin{pmatrix} \alpha _f \sqrt{\rho }\\ \pm \frac{\alpha _f c_f}{\sqrt{\rho }}\\ \mp \frac{\alpha _s c_s \beta _y}{\sqrt{\rho }}\\ \mp \frac{\alpha _s c_s \beta _z}{\sqrt{\rho }}\\ \alpha _f \frac{p_\parallel }{\sqrt{\rho }}\\ \alpha _f \sqrt{\rho } a^2\\ 0\\ \alpha _s a\beta _y\\ \alpha _s a\beta _z \end{pmatrix},{\tilde{R}}_{\lambda _{8,9}}= \frac{1}{2\sqrt{2}}\begin{pmatrix} \alpha _s \sqrt{\rho }\\ \pm \frac{\alpha _s c_s}{\sqrt{\rho }}\\ \pm \frac{\alpha _f c_f \beta _y}{\sqrt{\rho }}\\ \pm \frac{\alpha _f c_f \beta _z}{\sqrt{\rho }}\\ \alpha _s \frac{p_\parallel }{\sqrt{\rho }}\\ \alpha _s \sqrt{\rho } a^2\\ 0\\ -\alpha _f a\beta _y\\ -\alpha _f a\beta _z \end{pmatrix}, \end{aligned}$$

In y-direction, the entropy-scaled right eigenvectors corresponding to the above eigenvalues in the term of primitive variables are the following,

$$\begin{aligned}{\tilde{R}}_{\lambda _1}= & \begin{pmatrix} \frac{\sqrt{\rho }(2 p_\parallel + \rho )}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ 0 \end{pmatrix},{\tilde{R}}_{\lambda _2}= \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \sqrt{\frac{p_{\perp }}{2 \rho }}\\ 0 \end{pmatrix},{\tilde{R}}_{\lambda _3}= \begin{pmatrix} \frac{p_\parallel \sqrt{\rho }}{2 \sqrt{5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2}}\\ 0\\ 0\\ 0\\ \frac{p_\parallel (5 p_\parallel + 2 \rho )}{2 \sqrt{\rho (5 p_\parallel ^{2} + 4 p_\parallel \rho + \rho ^2)}}\\ 0\\ 0\\ 0\\ 0 \end{pmatrix},\\ {\tilde{R}}_{\lambda _{4,5}}= & \frac{1}{2}\begin{pmatrix} 0\\ \pm \frac{\sqrt{p_{\perp }}}{\rho }{\beta _z}\\ 0\\ \mp \frac{\sqrt{p_{\perp }}}{\rho }{\beta _x}\\ 0\\ 0\\ -\sqrt{\frac{p_{\perp }}{\rho }}{\beta _z}\\ 0\\ \sqrt{\frac{p_{\perp }}{\rho }}{\beta _x}\\ \end{pmatrix},{\tilde{R}}_{\lambda _{6,7}}= \frac{1}{2\sqrt{2}}\begin{pmatrix} \alpha _f \sqrt{\rho }\\ \mp \frac{\alpha _s c_s \beta _x}{\sqrt{\rho }}\\ \pm \frac{\alpha _f c_f}{\sqrt{\rho }}\\ \mp \frac{\alpha _s c_s \beta _z}{\sqrt{\rho }}\\ \alpha _f \frac{p_\parallel }{\sqrt{\rho }}\\ \alpha _f \sqrt{\rho } a^2\\ \alpha _s a\beta _x\\ 0\\ \alpha _s a\beta _z \end{pmatrix},{\tilde{R}}_{\lambda _{8,9}}= \frac{1}{2\sqrt{2}}\begin{pmatrix} \alpha _s \sqrt{\rho }\\ \pm \frac{\alpha _f c_f \beta _x}{\sqrt{\rho }}\\ \pm \frac{\alpha _s c_s}{\sqrt{\rho }}\\ \pm \frac{\alpha _f c_f \beta _z}{\sqrt{\rho }}\\ \alpha _s \frac{p_\parallel }{\sqrt{\rho }}\\ \alpha _s \sqrt{\rho } a^2\\ -\alpha _f a\beta _x\\ 0\\ -\alpha _f a\beta _z \end{pmatrix}, \end{aligned}$$

Proof of ${{\textbf {V}}}^\top \cdot {{\textbf {C}}}_x({{\textbf {U}}}) = {{\textbf {V}}}^\top \cdot {{\textbf {C}}}_y({{\textbf {U}}}) = 0$

The entropy variable ${{\textbf {V}}}:=\frac{\partial {\mathcal {E}}}{\partial {{\textbf {U}}}}$ is given by

$$\begin{aligned} {{\textbf {V}}}&=\Bigg \{5-s-\beta _\perp {{\textbf {u}}}^2,~2\beta _\perp {{\textbf {u}}},~-\beta _\parallel +\beta _\perp ,~-2\beta _\perp ,~2\beta _\perp \varvec{B}\Bigg \}^\top \end{aligned}$$

where $\beta _\perp =\frac{\rho }{p_{\perp }},$ and $\beta _\parallel =\frac{\rho }{p_\parallel }$. Now the dot product of ${{\textbf {V}}}^\top$ and first coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0,-\frac{b_x^2 {{\textbf {u}}}^2}{2},-\frac{b_xb_y {{\textbf {u}}}^2}{2},-\frac{b_xb_z {{\textbf {u}}}^2}{2},-\frac{2p_\parallel b_x}{\rho }({\varvec{b}}\cdot {{\textbf {u}}}), \Upsilon _1^x,0,0,0\right\} =\\&\quad = \left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( -\frac{b_x^2 {{\textbf {u}}}^2}{2}\right) +\left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( -\frac{b_xb_y {{\textbf {u}}}^2}{2}\right) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( -\frac{b_xb_z {{\textbf {u}}}^2}{2}\right) \\&\qquad +\left( \frac{\rho (p_\parallel -p_{\perp })}{p_\parallel p_{\perp }}\right) \left( -\frac{2p_\parallel b_x}{\rho }({\varvec{b}}\cdot {{\textbf {u}}})\right) +\left( -2\frac{\rho }{p_{\perp }}\right) (\Upsilon _1^x)\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( -\frac{b_x^2 {{\textbf {u}}}^2}{2}\right) +\left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( -\frac{b_xb_y {{\textbf {u}}}^2}{2}\right) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( -\frac{b_xb_z {{\textbf {u}}}^2}{2}\right) \\&\qquad +\left( \frac{\rho (p_\parallel -p_{\perp })}{p_\parallel p_{\perp }}\right) \left( -\frac{2p_\parallel b_x}{\rho }({\varvec{b}}\cdot {{\textbf {u}}})\right) \\&\qquad +\left( -2\frac{\rho }{p_{\perp }}\right) \left( -b_x(b\cdot {{\textbf {u}}})\frac{{{\textbf {u}}}^2}{2}-\frac{(p_\parallel - p_{\perp })b_x({\varvec{b}}\cdot {{\textbf {u}}})}{\rho }\right) \\&\quad =0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and second coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0,b_x^2 u_x,b_xb_y u_x,b_xb_z u_x,\frac{2p_\parallel b_x}{\rho }b_x,\Upsilon _2^x,0,0,0\right\} =\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) (b_x^2 u_x) + \left( 2\frac{\rho }{p_{\perp }} u_y\right) (b_xb_y u_x) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) (b_xb_z u_x) \\&\qquad + \left( \frac{\rho (p_\parallel -p_{\perp })}{p_\parallel p_{\perp }}\right) \left( \frac{2p_\parallel b_x}{\rho }b_x\right) +\left( -2\frac{\rho }{p_{\perp }}\right) \left( b_x({\varvec{b}}\cdot {{\textbf {u}}})u_x+\frac{(p_\parallel -p_{\perp })b^2_x}{\rho }\right) \\&\quad =0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and third coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0,b_x^2 u_y,b_xb_y u_y,b_xb_z u_y,\frac{2p_\parallel b_x}{\rho }b_y,\Upsilon _3^x,0,0,0\right\} =\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) (b_x^2 u_y) + \left( 2\frac{\rho }{p_{\perp }} u_y\right) (b_xb_y u_y) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) (b_xb_z u_y) \\&\qquad + \left( \frac{\rho (p_\parallel -p_{\perp })}{p_\parallel p_{\perp }}\right) \left( \frac{2p_\parallel b_x}{\rho }b_y\right) + \left( -2\frac{\rho }{p_{\perp }}\right) \left( b_x({\varvec{b}}\cdot {{\textbf {u}}})u_y+\frac{(p_\parallel - p_{\perp })b_xb_y}{\rho }\right) \\&\quad =0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and fourth coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0,b_x^2 u_z,b_xb_y u_z,b_xb_z u_z,\frac{2p_\parallel b_x}{\rho }b_z,\Upsilon _4^x,0,0,0\right\} =\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) (b_x^2 u_z) + \left( 2\frac{\rho }{p_{\perp }} u_y\right) (b_xb_y u_z) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) (b_xb_z u_z) \\&\qquad + \left( \frac{\rho (p_\parallel -p_{\perp })}{p_\parallel p_{\perp }}\right) \left( \frac{2p_\parallel b_x}{\rho }b_z\right) + \left( -2\frac{\rho }{p_{\perp }}\right) \left( b_x({\varvec{b}}\cdot {{\textbf {u}}})u_z+\frac{(p_\parallel - p_{\perp })b_xb_z}{\rho }\right) \\&\quad =0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and fifth coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0, \frac{3}{2}b_x^2, \frac{3}{2}b_xb_y, \frac{3}{2}b_xb_z ,0, \frac{3}{2}b_x(b\cdot {{\textbf {u}}}),0,0,0\right\} =\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( \frac{3}{2}b_x^2\right) + \left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( \frac{3}{2}b_xb_y\right) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( \frac{3}{2}b_xb_z\right) \\&\qquad + \left( -2\frac{\rho }{p_{\perp }}\right) \left( \frac{3}{2}b_x({\varvec{b}}\cdot {{\textbf {u}}})\right) \\&\quad =0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and sixth coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned}&{{\textbf {V}}}^\top \cdot \left\{ 0, -b_x^2, -b_xb_y, -b_xb_z ,0, -b_x(b\cdot {{\textbf {u}}}),0,0,0\right\} =\\&\quad =\left( 2\frac{\rho }{p_{\perp }} u_x\right) (-b_x^2) + \left( 2\frac{\rho }{p_{\perp }} u_y\right) (-b_xb_y) +\left( 2\frac{\rho }{p_{\perp }}u_z\right) (-b_xb_z)\\&\qquad + \left( -2\frac{\rho }{p_{\perp }}\right) \left( -b_x({\varvec{b}}\cdot {{\textbf {u}}})\right) \\&\quad =0 \end{aligned}$$

The seventh coloum, 8th coloum and ninth coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ denoted by $C_7,~C_8$ and $C_9$ is given below,

$$\begin{aligned}C_7= & \begin{pmatrix} 0 \\ b_x^2 B_x+\frac{2\Gamma _xb_x}{|\varvec{B}|} \\ b_xb_y B_x+\frac{\Gamma _xb_y - \Delta Pb_{yxx}}{|\varvec{B}|} \\ b_xb_z B_x+\frac{\Gamma _xb_z - \Delta Pb{zxx}}{|\varvec{B}|} \\ 0 \\ b_x({\varvec{b}}\cdot {{\textbf {u}}})B_x + \Theta ^x_1 \\ 0 \\ 0 \\ 0 \end{pmatrix},C_8= \begin{pmatrix} 0 \\ b_x^2 B_y - \frac{2\Delta Pb_{yxx}}{|\varvec{B}|} \\ b_x b_y B_y+\frac{\Gamma _y b_x - \Delta Pb_{xyy}}{|\varvec{B}|} \\ b_xb_z B_y-\frac{2\Delta Pb_{xyz}}{|\varvec{B}|} \\ 0 \\ b_x({\varvec{b}}\cdot {{\textbf {u}}})B_y + \Theta ^x_2 \\ 0 \\ 0 \\ 0 \end{pmatrix},\\ C_9= & \begin{pmatrix} 0 \\ b_x^2 B_z-\frac{2\Delta Pb_{zxx}}{|\varvec{B}|} \\ b_xb_y B_z-\frac{2\Delta Pb_{xyz}}{|\varvec{B}|} \\ b_xb_z B_z + \frac{\Gamma _zb_x - \Delta Pb_{xzz}}{|\varvec{B}|} \\ 0 \\ b_x({\varvec{b}}\cdot {{\textbf {u}}})B_z + \Theta ^x_3 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and seventh coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned} {{\textbf {V}}}^\top \cdot C_7&= \left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( b_x^2 B_x+\frac{2(p_\parallel - p_{\perp })b_x(1-b_x^2)}{|\varvec{B}|}\right) \\&\quad + \left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( b_xb_y B_x+\frac{(p_\parallel - p_{\perp })b_y(1-b_x^2)}{|\varvec{B}|}- \frac{(p_\parallel - p_{\perp })b_yb_x^2}{|\varvec{B}|}\right) \\&\quad +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( b_xb_z B_x+\frac{(p_\parallel - p_{\perp })b_z(1-b_x^2)}{|\varvec{B}|} - \frac{(p_\parallel - p_{\perp })b_zb_x^2}{|\varvec{B}|}\right) \\&\quad +\left( -2\frac{\rho }{p_{\perp }}\right) \Bigg (b_x({\varvec{b}}\cdot {{\textbf {u}}})B_x + \left\{ (p_\parallel - p_{\perp })({\varvec{b}}\cdot {{\textbf {u}}})+(p_\parallel - p_{\perp })b_x u_x\right\} \Bigg (\frac{1-b^2_x}{|\varvec{B}|}\Bigg )\\&\quad -(p_\parallel - p_{\perp })\frac{b^2_x b_y u_y}{|\varvec{B}|}- (p_\parallel - p_{\perp })\frac{b^2_x b_z u_z}{|\varvec{B}|}\Bigg )\\&=0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and 8th coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned} {{\textbf {V}}}^\top \cdot C_8&= \left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( b_x^2 B_y-\frac{2(p_\parallel - p_{\perp })b_yb_x^2}{|\varvec{B}|}\right) \\&\quad + \left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( b_xb_y B_y+\frac{(p_\parallel - p_{\perp })b_x(1-b_y^2)}{|\varvec{B}|} - \frac{(p_\parallel - p_{\perp })b_xb_y^2}{|\varvec{B}|}\right) \\&\quad +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( b_xb_z B_y-\frac{2(p_\parallel - p_{\perp })b_xb_yb_z}{|\varvec{B}|}\right) \\&\quad +\left( -2\frac{\rho }{p_{\perp }}\right) \Bigg (b_x({\varvec{b}}\cdot {{\textbf {u}}})B_y + (p_\parallel - p_{\perp })b_x u_y \Bigg (\frac{1-b^2_y}{|\varvec{B}|}\Bigg ) - (p_\parallel - p_{\perp })\frac{b_x b_y b_z u_z}{|\varvec{B}|} \\&\quad -\{(p_\parallel - p_{\perp })({\varvec{b}}\cdot {{\textbf {u}}})+(p_\parallel - p_{\perp })b_x u_x\}\frac{b_x b_y}{|\varvec{B}|}\Bigg )\\&=0 \end{aligned}$$

Now the dot product of ${{\textbf {V}}}^\top$ and ninth coloum of ${{\textbf {C}}}_x({{\textbf {U}}})$ is:

$$\begin{aligned} {{\textbf {V}}}^\top \cdot C_9&= \left( 2\frac{\rho }{p_{\perp }} u_x\right) \left( b_x^2 B_z-\frac{2(p_\parallel - p_{\perp })b_zb_x^2}{|\varvec{B}|}\right) \\&\quad + \left( 2\frac{\rho }{p_{\perp }} u_y\right) \left( b_xb_y B_z-\frac{2(p_\parallel - p_{\perp })b_xb_yb_z}{|\varvec{B}|}\right) \\&\quad +\left( 2\frac{\rho }{p_{\perp }}u_z\right) \left( b_xb_z B_z+\frac{(p_\parallel - p_{\perp })b_x(1-b_z^2)}{|\varvec{B}|} - \frac{(p_\parallel - p_{\perp })b_xb_z^2}{|\varvec{B}|}\right) \\&\quad +\left( -2\frac{\rho }{p_{\perp }}\right) \Bigg (b_x({\varvec{b}}\cdot {{\textbf {u}}})B_z + (p_\parallel - p_{\perp })b_x u_z \Bigg (\frac{1-b^2_z}{|\varvec{B}|}\Bigg ) - (p_\parallel - p_{\perp })\frac{b_x b_y b_z u_y}{|\varvec{B}|}\\&\quad -\{(p_\parallel - p_{\perp })({\varvec{b}}\cdot {{\textbf {u}}})+(p_\parallel - p_{\perp })b_x u_x\}\frac{b_x b_z}{|\varvec{B}|}\Bigg )\\&=0 \end{aligned}$$

Similarly we can proof that ${{\textbf {V}}}^\top \cdot {{\textbf {C}}}_y({{\textbf {U}}}) = 0$ for y-direction.

Non-Symmetrizability of CGL System

In this section, we have discussed the symmetrizability of the following CGL equations. Consider CGL equations in one dimension as follows:

$$\begin{aligned} \frac{\partial {{\textbf {U}}}}{\partial t}+\frac{\partial {{\textbf {f}}}_{x}}{\partial x} + {{\textbf {C}}}_{x}({{\textbf {U}}})\frac{\partial {{\textbf {U}}}}{\partial x} =0. \end{aligned}$$

(62)

where ${{\textbf {U}}},~{{\textbf {f}}}_{x}$ and ${{\textbf {C}}}_{x}({{\textbf {U}}})$ are defined in (3). Follow [68], the CGL equations are said to be symmetrizable if the change of variable ${{\textbf {U}}}\rightarrow {{\textbf {V}}}$ applied to (62) and can be written as:

$$\begin{aligned} \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {V}}}}\frac{\partial {{\textbf {V}}}}{\partial {t}}+\left( \frac{\partial {{\textbf {f}}}_x}{\partial {{\textbf {U}}}} +{{\textbf {C}}}_{x}({{\textbf {U}}}) \right) \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {V}}}}\frac{\partial {{\textbf {V}}}}{\partial {x}}=0. \end{aligned}$$

the matrix $\frac{\partial {{\textbf {U}}}}{\partial {{\textbf {V}}}}$ is a symmetric, positive definite matrix and $\aleph ({{\textbf {U}}}) = \left( \frac{\partial {{\textbf {f}}}_x}{\partial {{\textbf {U}}}} +{{\textbf {C}}}_{x}({{\textbf {U}}}) \right) \frac{\partial {{\textbf {U}}}}{\partial {{\textbf {V}}}}$ is symmetric matrix. For the CGL equation (62), we calculate matrix $\aleph ({{\textbf {U}}})$ to examine its symmetry and conclude that CGL equations are non-symmetrizable. The resultant matrix $\aleph ({{\textbf {U}}}) -\aleph ({{\textbf {U}}})^\top$ is given below:

$$\begin{aligned} \begin{pmatrix} 0 & -\frac{\Delta Pb_x^2}{2} & -\frac{\Delta Pb_x b_y}{2} & -\frac{\Delta Pb_x b_z}{2} & 0 & -\frac{\Delta Pb_x ({\varvec{b}}\cdot {{\textbf {u}}})}{2} & 0 & 0 & 0\\ \frac{\Delta Pb_x^2}{2} & 0 & \frac{\Delta Pb_x (b_x u_y - b_y u_x)}{2} & \frac{\Delta Pb_x (b_x u_z - b_z u_x)}{2} & \frac{3 \Delta Pp_\parallel b_x^2}{2\rho } & -\frac{\alpha _{xyz}}{4\rho } & -\frac{p_{\perp }\omega _1}{2\rho } & -\frac{\Delta Pp_{\perp }b_y b_x^2}{\rho |\varvec{B}|} & -\frac{\Delta Pp_{\perp }b_z b_x^2}{\rho |\varvec{B}|}\\ \frac{\Delta Pb_x b_y}{2} & \frac{\Delta Pb_x (b_y u_x - b_x u_y)}{2} & 0 & \frac{\Delta Pb_x (b_y u_z - b_z u_y)}{2} & \frac{3 \Delta Pp_\parallel b_x b_y}{2\rho } & -\frac{\alpha _{yxz}}{4\rho } & -\frac{p_{\perp }\omega _2}{2\rho } & \frac{\Delta Pp_{\perp }b_x (1-2b_y^2)}{2\rho |\varvec{B}|} & -\frac{\Delta Pp_{\perp }b_x b_y b_z}{\rho |\varvec{B}|}\\ \frac{\Delta Pb_x b_z}{2} & \frac{\Delta Pb_x (b_z u_x - b_x u_z)}{2} & \frac{\Delta Pb_x (b_z u_y - b_y u_z)}{2} & 0 & \frac{3 \Delta Pp_\parallel b_x b_z}{2\rho } & -\frac{\alpha _{zxy}}{4\rho } & -\frac{p_{\perp }\omega _3}{2\rho } & -\frac{\Delta Pp_{\perp }b_x b_y b_z}{\rho |\varvec{B}|} & \frac{\Delta Pp_{\perp }b_x (1-2b_z^2)}{2\rho |\varvec{B}|} \\ 0 & -\frac{3 \Delta Pp_\parallel b_x^2}{2\rho } & -\frac{3 \Delta Pp_\parallel b_x b_y}{2\rho } & -\frac{3 \Delta Pp_\parallel b_x b_z}{2\rho } & 0 & -\frac{3 \Delta Pp_\parallel b_x({\varvec{b}}\cdot {{\textbf {u}}}) }{2\rho } & 0 & 0 & 0 \\ \frac{\Delta Pb_x ({\varvec{b}}\cdot {{\textbf {u}}})}{2} & \frac{\alpha _{xyz}}{4\rho } & \frac{\alpha _{yxz}}{4\rho } & \frac{\alpha _{zxy}}{4\rho } & \frac{3 \Delta Pp_\parallel b_x({\varvec{b}}\cdot {{\textbf {u}}}) }{2\rho } & 0 & -\frac{p_{\perp }\beta _1}{2 \rho } & \frac{p_{\perp }\beta _2}{2 \rho } & \frac{p_{\perp }\beta _3}{2 \rho }\\ 0 & \frac{p_{\perp }\omega _1}{2\rho } & \frac{p_{\perp }\omega _2}{2\rho } & \frac{p_{\perp }\omega _3}{2\rho } & 0 & \frac{p_{\perp }\beta _1}{2 \rho } & 0 & \frac{p_{\perp }u_y}{2\rho } & \frac{p_{\perp }u_z}{2\rho }\\ 0 & \frac{\Delta Pp_{\perp }b_y b_x^2}{\rho |\varvec{B}|} & -\frac{\Delta Pp_{\perp }b_x (1-2b_y^2)}{2\rho |\varvec{B}|} & \frac{\Delta Pp_{\perp }b_x b_y b_z}{\rho |\varvec{B}|} & 0 & -\frac{p_{\perp }\beta _2}{2 \rho } & -\frac{p_{\perp }u_y}{2\rho } & 0 & 0\\ 0 & \frac{\Delta Pp_{\perp }b_z b_x^2}{\rho |\varvec{B}|} & \frac{\Delta Pp_{\perp }b_x b_y b_z}{\rho |\varvec{B}|} & -\frac{\Delta Pp_{\perp }b_x (1-2b_z^2)}{2\rho |\varvec{B}|} & 0 & -\frac{p_{\perp }\beta _3}{2 \rho } & -\frac{p_{\perp }u_z}{2\rho } & 0 & 0 \end{pmatrix}. \end{aligned}$$

where, $\Delta P= (p_\parallel -p_{\perp })$ and $\alpha _{lmn}= 2 p_{\perp }B_x B_l + \Delta Pb_x\Big (2 \rho u_l (b_m u_m + b_n u_n) - b_l$ $(3p_\parallel +2p_{\perp }+ \rho (u_m^2 +u_n^2 - u_l^2))\Big );~~\forall ~l,m,n \in \{x,y,z\}$

$\beta _1 = B_y u_y + B_z u_z + \frac{\Delta P\left( -b_y u_y - b_z u_z + 2 b_x(({\varvec{b}}\cdot {{\textbf {u}}})b_x-u_x)\right) }{|\varvec{B}|}$ and $\beta _l = B_x u_l - \frac{\Delta Pb_x(2 b_l({\varvec{b}}\cdot {{\textbf {u}}})-u_l)}{|\varvec{B}|};~~\forall ~l\in \{y,z\}$

$\omega _1 = B_x - \frac{2 \Delta Pb_x (1-b_x^2)}{|\varvec{B}|}$ and $\omega _l = B_l - \frac{\Delta Pb_l (1 - 2 b_x^2)}{|\varvec{B}|};~~\forall ~l\in \{y,z\}$