Entropy Stable Discontinuous Galerkin Schemes for Two-Fluid Relativistic Plasma Flow Equations

Abstract

This article proposes entropy stable discontinuous Galerkin schemes (DG) for two-fluid relativistic plasma flow equations. These equations couple the flow of relativistic fluids via electromagnetic quantities evolved using Maxwell’s equations. The proposed schemes are based on the Gauss–Lobatto quadrature rule, which has the summation by parts property. We exploit the structure of the equations having the flux with three independent parts coupled via nonlinear source terms. We design entropy stable DG schemes for each flux part, coupled with the fact that the source terms do not affect entropy, resulting in an entropy stable scheme for the complete system. The proposed schemes are then tested on various test problems in one and two dimensions to demonstrate their accuracy and stability.

Appendices

A Computation of Primitive Variables from the Conservative Variables

The evolutionary equations for the two-fluid relativistic plasma system (1) relies on the conserved quantities, $D_\alpha$, ${\varvec{M}}_\alpha$ and ${\mathcal {E}}_\alpha$ and primitive quantities, $\rho _\alpha$, ${\varvec{v}}_\alpha$ and $p_\alpha$. We use the numerical procedure given in [11, 59] for the extraction of primitive variables from the conservative variables. As the procedure for obtaining primitives for the ions and electron fluid quantities is identical, we only focus on the ion-variables. Accordingly, for simplicity, we suppress the subscript $\alpha$. For the ideal equation of state (2), we obtain the following quartic polynomial for the velocity variable $v=|{\varvec{v}}|$ with real coefficients which only depend on the conservative variables.

$$\begin{aligned} {v}^4 + c_3 {v}^3 + c_2 {v}^2 + c_1 {v} + c_0=0, \end{aligned}$$

(45)

where

$$\begin{aligned} c_3 = -\frac{2 \gamma (\gamma - 1) M {\mathcal {E}}}{(\gamma - 1)^2({M}^2 + D^2)},\;\; c_2= \frac{(\gamma ^2 {\mathcal {E}}^2 + 2(\gamma - 1){M}^2 - (\gamma - 1)^2 D^2)}{(\gamma - 1)^2({M}^2 + D^2)}, \\ c_1=\frac{-2 \gamma {M} {\mathcal {E}}}{(\gamma - 1)^2({M}^2 + D^2)},\;\; c_0 = \frac{{M}^2}{(\gamma - 1)^2({M}^2 + D^2)}, \end{aligned}$$

with $M=(M_x^2+M_y^2+M_z^2)^{1/2}$. The quartic polynomial (45) is solved using the Newton’s root solver with the number of iterations fixed to ten. We use the following initial guess $v_0$ to initialize the root solver,

$$\begin{aligned} v_0=\frac{1}{2}(v_{lb}+v_{ub})+z, \end{aligned}$$

where

$$\begin{aligned} v_{lb}&=\frac{1}{2M(\gamma - 1)}(\gamma {\mathcal {E}} - \sqrt{\gamma ^2 {\mathcal {E}}^2 - 4(\gamma - 1)M^2} ), \\ v_{ub}&=\min {\left( 1, \frac{M}{{\mathcal {E}}}+ \delta \right) }, \\ z&= {\left\{ \begin{array}{ll} \dfrac{1}{2}\left( 1-\dfrac{D}{{\mathcal {E}}}\right) (u_{lb}-u_{ub}) &{} \text { if } u_{lb}>10^{-9} \\ 0 &{} \text { otherwise} \end{array}\right. }. \end{aligned}$$

After obtaining the velocity norm, v, we can extract the primitive variables using the following expressions.

$$\begin{aligned}{} & {} \rho = \frac{D}{\Gamma }, \\{} & {} v_x=\frac{M_x}{M} v, \ \ \ v_y=\frac{M_y}{M} v, \ \ \ v_z=\frac{M_z}{M} v, \\{} & {} p=(\gamma -1)({\mathcal {E}}- M_x v_x- M_y v_y -M_z v_z -\rho ). \end{aligned}$$

B Right Eigenvectors for the Two-Dimensional Two-Fluid Relativistic Plasma System

In this appendix, we provide expressions of the right eigenvectors for the two-dimensional relativistic plasma system (4). The set of right eigenvectors of the matrix ${\textbf{A}}^x$ corresponding to the eigenvalues $\mathbf {\Lambda }^x$ of the system (4) is obtained by setting $d=x$ in the ordered set ${\textbf{R}}^d_{\Lambda ^d} = \left\{ \left( {\textbf{R}}_{\Lambda ^d}^d \right) _n: \ n = 1, \ 2,\ 3,\dots , 18 \right\}$. For $n=1,2,3,\dots ,18,$ the vectors $\left( {\textbf{R}}_{\Lambda ^d}^d\right) _n$ are given by

$$\begin{aligned} \left( {\textbf{R}}_{\Lambda ^d}^d\right) _n = {\left\{ \begin{array}{ll} \Big (({\textbf{R}}_{i,k}^{d})_{1 \times 5}, {\textbf{0}}_{1 \times 5}, {\textbf{0}}_{1 \times 8}\Big )^\top , &{} 1 \le n \le 5, \ k = n \\ \Big ({\textbf{0}}_{1 \times 5}, ({\textbf{R}}_{e,k}^{d})_{1 \times 5}, {\textbf{0}}_{1 \times 8}\Big )^\top , &{} 6 \le n \le 10, \ k = n-5 \\ \Big ({\textbf{0}}_{1 \times 5}, {\textbf{0}}_{1 \times 5}, ({\textbf{R}}_{m,k}^{d})_{1 \times 8}\Big )^\top , &{} 11 \le n \le 18, \ k = n-10, \end{array}\right. } \end{aligned}$$

(46)

where ${\textbf{R}}^{d}_{\alpha ,k}$, $\alpha \in \{i,e\}$, is the $k^{th}$ column vector of the $5 \times 5$ right eigenvector matrices ${\textbf{R}}^d_\alpha$ of the flux jacobians $\dfrac{\partial {\textbf{f}}^d_\alpha }{\partial {\textbf{U}}_\alpha }$, and ${\textbf{R}}^{d}_{m,k}$ is the $k^{th}$ column vector of the right eigenvector matrix ${\textbf{R}}^d_m$ of the flux jacobian matrix $\dfrac{\partial {\textbf{f}}^d_m}{\partial {\textbf{U}}_m}$. The matrices ${\textbf{R}}^{k}_{\alpha }$ and ${\textbf{R}}^{k}_{m}$ has the following expressions.

• For $d =x,y$, $\alpha \in \{i,e\}$, the right eigenvector matrix ${\textbf{R}}^d_\alpha$ is given by the equation

  $$\begin{aligned} {\textbf{R}}_{\alpha }^{d}= \left( \dfrac{\partial {\textbf{U}}_\alpha }{\partial {\textbf{W}}_\alpha } \right) {\textbf{R}}_{\alpha ,{\textbf{W}}}^d \end{aligned}$$

  where ${\textbf{R}}_{\alpha ,{\textbf{W}}}^d$ is the matrix of right eigenvectors of system (4) written in primitive form. For $d=x$, the matrix ${\textbf{R}}_{\alpha ,{\textbf{W}}}^d$ has the expression

  $$\begin{aligned} {\textbf{R}}_{\alpha ,{\textbf{W}}}^x = \begin{pmatrix} \frac{1}{c_\alpha ^2 h_\alpha } &{} 1 &{} 0 &{} 0 &{} \frac{1}{c_\alpha ^2 h_\alpha } \\ \frac{-\sqrt{Q^x_\alpha }}{c_\alpha h_\alpha \Gamma _\alpha \rho _\alpha } &{} 0 &{} 0 &{} 0 &{} \frac{+\sqrt{Q^x_\alpha }}{c_\alpha h_\alpha \Gamma _\alpha \rho _\alpha } \\ \frac{\left( c_\alpha -\Gamma _\alpha \sqrt{Q^x_\alpha } {v_{x_\alpha }}\right) {v_{y_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{x_\alpha }^2-1\right) } &{} 0 &{} 1 &{} 0 &{} \frac{\left( c_\alpha +\Gamma _\alpha \sqrt{Q^x_\alpha } {v_{x_\alpha }}\right) {v_{y_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( u_{x_\alpha }^2-1\right) } \\ \frac{\left( c_\alpha -\Gamma _\alpha \sqrt{Q^x_\alpha } {v_{x_\alpha }}\right) {v_{z_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{x_\alpha }^2-1\right) } &{} 0 &{} 1 &{} 0 &{} \frac{\left( c_\alpha +\Gamma _\alpha \sqrt{Q^x_\alpha } {v_{x_\alpha }}\right) {v_{z_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{x_\alpha }^2-1\right) } \\ 1 &{} 0 &{} 0 &{} 0 &{} 1 \end{pmatrix}. \end{aligned}$$

• The eigenvector matrix ${\textbf{R}}^{d}_{m}$ for $d=y$ is given by

  $$\begin{aligned} {\textbf{R}}_{m}^{x}= \begin{pmatrix} 0 &{}-1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{}-1 &{} 0 &{} 0 \\ 0 &{} 0 &{}-1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{pmatrix}. \end{aligned}$$

Remark 4

For the $y-$directional flux, ${\textbf{f}}^y$, we proceed similarly to obtain the set of eigenvalues $\mathbf {\Lambda }^y$ of the jacobian matrix $\dfrac{\partial {\textbf{f}}^y}{\partial {\textbf{U}}}$ as

$$\begin{aligned} \mathbf {\Lambda }^y = \biggl \{&\frac{(1-c_i^2)v_{y_i}-(c_i/\Gamma _i) \sqrt{Q_i^y}}{1-c_i^2 |{\varvec{v}}|_i^2},\ v_{y_i}, \ v_{y_i}, \ v_{y_i}, \frac{(1-c_i^2)v_{y_i}+(c_i/\Gamma _i) \sqrt{Q_i^y}}{1-c_i^2 |{\varvec{v}}|_i^2}, \\&\frac{(1-c_e^2)v_{y_e}-(c_e/\Gamma _e) \sqrt{Q_e^y}}{1-c_e^2 |{\varvec{v}}|_e^2},\ v_{y_e}, \ v_{y_e}, \ v_{y_e}, \frac{(1-c_e^2)v_{y_e}+(c_e/\Gamma _e) \sqrt{Q_e^y}}{1-c_e^2 |{\varvec{v}}|_e^2}, \\&-\chi , \ -\kappa , \ -1, \ -1, \ 1, \ 1, \ \kappa , \ \chi \biggr \}, \end{aligned}$$

where, $Q_\alpha ^y=1-u_{y_\alpha }^2-c_\alpha ^2 (u_{x_\alpha }^2+u_{z_\alpha }^2), \ \alpha \in \{ i, \ e \}$. The corresponding right eigenvectors can be obtained by taking $d=y$ in the ordered set ${\textbf{R}}^d_{\Lambda ^d} = \left\{ \left( {\textbf{R}}_{\Lambda ^d}^d\right) _n: \ n = 1, \ 2,\ 3,\dots , \ 18\right\}$ where $\left( {\textbf{R}}_{\Lambda ^d}^d\right) _n$ is defined by Eq. (46) and the updated $y-$directional matrices ${\textbf{R}}_{\alpha ,{\textbf{W}}}^y$ and ${\textbf{R}}^{y}_{m}$ are given by

$$\begin{aligned} {\textbf{R}}_{\alpha ,{\textbf{W}}}^y = \begin{pmatrix} \frac{1}{c_\alpha ^2 h_\alpha } &{} 1 &{} 0 &{} 0 &{} \frac{1}{c_\alpha ^2 h_\alpha } \\ \frac{\left( c_\alpha -\Gamma _\alpha \sqrt{Q^y_\alpha } {v_{y_\alpha }}\right) {v_{x_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{y_\alpha }^2-1\right) } &{} 0 &{} 1 &{} 0 &{} \frac{\left( c_\alpha +\Gamma _\alpha \sqrt{Q^y_\alpha } {v_{y_\alpha }}\right) {v_{x_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{y_\alpha }^2-1\right) } \\ \frac{-\sqrt{Q^y_\alpha }}{c_\alpha h_\alpha \Gamma _\alpha \rho _\alpha } &{} 0 &{} 0 &{} 0 &{} \frac{+\sqrt{Q^y_\alpha }}{c_\alpha h_\alpha \Gamma _\alpha \rho _\alpha } \\ \frac{\left( c_\alpha -\Gamma _\alpha \sqrt{Q^y_\alpha } {v_{y_\alpha }}\right) {v_{z_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{y_\alpha }^2-1\right) } &{} 0 &{} 0 &{} 1 &{} \frac{\left( c_\alpha +\Gamma _\alpha \sqrt{Q^y_\alpha } {v_{y_\alpha }}\right) {v_{z_\alpha }}}{c_\alpha h_\alpha \Gamma ^2_\alpha \rho _\alpha \left( v_{y_\alpha }^2-1\right) } \\ 1 &{} 0 &{} 0 &{} 0 &{} 1 \end{pmatrix}, \\ {\textbf{R}}_{m}^{y}= \begin{pmatrix} 0 &{} 0 &{} 0 &{}-1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{}-1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{}-1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{pmatrix} \end{aligned}$$