Entropy-stable schemes for relativistic hydrodynamics equations

Abstract

In this article, we propose high-order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high-order entropy dissipative operators. We first design a higher-order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO-based sign-preserving reconstruction of scaled entropy variables, which results in higher-order entropy-stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes.

Appendices

Computation of primitive variables

The RHD equations involve the conserved quantities, D, $\varvec{m}$ and E; but to solve the RHD equations numerically, we need to know the explicit values of the primitive variables, $\rho$, $\mathbf{u}$ and p. The primitive variables can be extracted by inverting the RHD equations (1). For the ideal equation of state (4), [30] showed that (using Eq. (1)) the variable ${u} (=|\mathbf{u}|)$ satisfies

$$\begin{aligned} {u}^4 + a_3 {u}^3 + a_2 {u}^2 + a_1 {u} + a_0=0, \end{aligned}$$

(25)

where

$$\begin{aligned} a_3= & {} -\frac{2 \gamma (\gamma - 1) ME}{(\gamma - 1)^2({M}^2 + D^2)},\;\; a_2= \frac{(\gamma ^2 E^2 + 2(\gamma - 1){M}^2 - (\gamma - 1)^2 D^2)}{(\gamma - 1)^2({M}^2 + D^2)},\\ a_1= & {} \frac{-2 \gamma {M} E}{(\gamma - 1)^2({M}^2 + D^2)},\;\; a_0 = \frac{{M}^2}{(\gamma - 1)^2({M}^2 + D^2)}, \end{aligned}$$

with $M=(m_x^2+m_y^2+m_z^2)^{1/2}$. Equation (25) could either be solved using the analytical methods or using an approximate root solver. In [30], authors have suggested the following expressions for the lower bound $u_{\mathrm{lb}}$ and upper bound $u_{\mathrm{ub}}$:

$$\begin{aligned} u_{\mathrm{lb}}=\frac{1}{2M(\gamma - 1)}(\gamma E - \sqrt{\gamma ^2 E^2 - 4(\gamma - 1)M^2} ) \end{aligned}$$

and

$$\begin{aligned} u_{\mathrm{ub}}=\min {\left( 1, \frac{M}{E}+ \delta \right) }, \end{aligned}$$

where $\delta \approx 10^{-6}$. Furthermore, to find the root of Eq. (25) using the Newton–Raphson method, [30] suggested the initial guess $u_o$ as,

$$\begin{aligned} u_o=\frac{1}{2}(u_{\mathrm{lb}}+u_{\mathrm{ub}})+z, \end{aligned}$$

where $z=\frac{1}{2}\left( 1-\frac{D}{E}\right) (u_{\mathrm{lb}}-u_{\mathrm{ub}})$ if $u_{\mathrm{lb}}>10^{-9}$ and $z=0$ otherwise. We then obtain $u_o$ between $u_{\mathrm{lb}}$ and $u_{\mathrm{ub}}$. Also, an accuracy of nine digits can be achieved in at most five iterations of the Newton–Raphson root solver.

For the analytic solution, we can use the direct formulation of the roots of the quartic equations. The general form of analytical roots is available in [1]. We note that Eq. (25) has two complex roots and two real roots. Among two real roots, only one root lies between the lower and upper bounds suggested by [30], which is given by,

$$\begin{aligned} u = \frac{-B + \sqrt{B^2-4 C}}{2}, \end{aligned}$$

where,

$$\begin{aligned} \begin{aligned} B&=\frac{a_3}{2}+\sqrt{\frac{a_3^2}{4}+z_1-a_2}, \ C=\frac{z_1}{2}-\sqrt{\left( \frac{z_1}{2}\right) ^2-a_0}, \ z_1=(s_1+s_2)-\frac{b_{2}}{3}, \ s_1={(r+\sqrt{q^3+r^2})}^{1/3},\\ s_2&={(r-\sqrt{q^3+r^2})}^{1/3}, \ q=\frac{b_1}{3}-\frac{b_2^2}{9}, \ r=\frac{1}{6}(b_1 b_2-3 b_0)-\frac{1}{27} b_2^3,\ b_0=-(a_1^2+a_0 a_3^2-4 a_0 a_2), \\ b_1&=a_1 a_3-4 a_0 \ \text {and } b_2=-a_2. \end{aligned} \end{aligned}$$

Once we have the value of u, we can find all the primitive variables, i.e. $\rho ,\ u_i, \ p,$ using the expressions

$$\begin{aligned} \rho= & {} \frac{D}{\Gamma }, \\ u_x= & {} \frac{m_x}{M} u, \ \ \ u_y=\frac{m_y}{M} u, \ \ \ u_z=\frac{m_z}{M} u, \\ p= & {} (\gamma -1)(E- m_x u_x- m_y u_y -m_z u_z -\rho ). \end{aligned}$$

In this article, we use numerical approach to find the primitive variables from the conservative variables.

Entropy scaled right eigenvectors

In this section, we present the entropy scaled right eigenvectors. First, we consider the case of x-direction right eigenvectors. We rewrite the RHD system (3),

$$\begin{aligned} \frac{\partial \mathbf {U}}{\partial t}+ \mathbf {A}^x\frac{\partial \mathbf {U}}{\partial x } = 0, \text { where }\ \mathbf {A}^x=\frac{\partial \mathbf {f}^x}{\partial \mathbf {U}}. \end{aligned}$$

Following [29], the eigenvalues of the Jacobian matrix $\mathbf {A}^x$ are given by:

$$\begin{aligned} \lambda _1^x =\frac{(1-c^2)u_x-(c/\Gamma ) \sqrt{Q^x}}{1-c^2 \varvec{u}^2}, \ \ \lambda _2^x= \lambda _3^x =u_x \ \text {and} \ \ \lambda _4^x = \ \frac{(1-c^2)u_x+(c/\Gamma ) \sqrt{Q^x}}{1-c^2 \varvec{u}^2}. \end{aligned}$$

Here, $Q^x=1-u_x^2-c^2 u_y^2$ and c is the sound speed given by the expression

$$\begin{aligned} c^2=-\frac{\rho }{n h} \frac{\partial h}{\partial \rho }, \end{aligned}$$

where $n=k-1$ with $k = \frac{\gamma }{\gamma -1}$, for all $\mathbf {U} \in \Omega$. Note that $Q^x=1-u_x^2-c^2 u_y^2 > 0$; therefore, all the eigenvalues are real. A straightforward calculation results in the following expressions for the eigenvectors in primitive variables:

The eigenvector matrix for the conservative system, $\tilde{\mathbf {R}}^x$, can be derived using,

$$\begin{aligned} \tilde{\mathbf {R}}^{x}=\frac{\partial \mathbf {U}}{\partial \mathbf {w}} \tilde{\mathbf {R}}_w^{x}, \end{aligned}$$

(26)

where $\tilde{\mathbf {R}}_w^{x} =(e_1, \ e_2, \ e_3,\ e_4)$, i.e. the matrix of right eigenvectors for the primitive system. To obtain the scaled right eigenvector matrix ${{\mathbf {R}}}^x$, we need to find a scaling matrix $\mathbf {T}^x$ such that the scaled right eigenvector matrix, ${{\mathbf {R}}}^x= \tilde{\mathbf {R}}^x \mathbf {T}^x$, satisfies

$$\begin{aligned} \frac{\partial \mathbf {U}}{\partial \mathbf {v}} ={{\mathbf {R}}^x} {{}{{\mathbf {R}}^x}}^\top . \end{aligned}$$

(27)

We first derive the expression of the entropy scaled right eigenvector for the primitive variable system, ${\mathbf {R}}^x_w$. Following the procedure of [4, 31], the scaling matrix $\mathbf {T}^x$ is the square root of the matrix $\mathbf {Y}^x$, where

$$\begin{aligned} \mathbf {Y}^x= {{}\tilde{\mathbf {R}}^x_w}^{-1} \frac{\partial \mathbf {w}}{\partial \mathbf{v} } \frac{\partial \mathbf {U}}{\partial \mathbf {w}}^{- \top } {{}\tilde{\mathbf {R}}^x_w}^{- \top }, \end{aligned}$$

which on a long calculation results in,

$$\begin{aligned} \mathbf {Y}^x= \begin{pmatrix} \frac{c^2 h p \left( 1+\frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) } &{} 0 &{} 0 &{} 0\\ 0 &{} \frac{\rho }{k \Gamma } &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{p}{h \Gamma ^5 \rho ^2 \left( 1-{u_x}^2\right) } &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{c^2 h p \left( 1- \frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) } \end{pmatrix}. \end{aligned}$$

Consequently,

$$\begin{aligned} \mathbf {T}^x=\sqrt{ \mathbf {Y}^x}= \begin{pmatrix} \sqrt{ \frac{c^2 h p \left( 1+\frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }} &{} 0 &{} 0 &{} 0 \\ 0 &{} \sqrt{\frac{\rho }{k \Gamma }} &{} 0 &{} 0 \\ 0 &{} 0 &{} \sqrt{ \frac{p}{h \Gamma ^5 \rho ^2 \left( 1-{u_x}^2\right) }} &{} 0 \\ 0 &{} 0 &{} 0 &{} \sqrt{ \frac{c^2 h p \left( 1- \frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }} \end{pmatrix}. \end{aligned}$$

So, the entropy scaled right eigenvector matrix for the primitive system is given by

$$\begin{aligned} {{\mathbf {{R}}}^x_w}&= {\tilde{\mathbf {{R}}}^x_w} \mathbf {T}^x\\&= \begin{pmatrix} \frac{A^x_1}{c^2 h} &{} A^x_2 &{} 0 &{} \frac{A^x_4}{c^2 h} \\ \frac{-A^x_1\sqrt{Q^x}}{c h \Gamma \rho } &{} 0 &{} 0 &{} \frac{A^x_4 \sqrt{Q^x}}{c h \Gamma \rho } \\ \frac{A^x_1\left( c-\Gamma \sqrt{Q^x} {u_x}\right) {u_y}}{c h \Gamma ^2 \rho \left( {u_x}^2-1\right) } &{} 0 &{} A^x_3 &{} \frac{A^x_4\left( c+\Gamma \sqrt{Q^x} {u_x}\right) {u_y}}{c h \Gamma ^2 \rho \left( {u_x}^2-1 \right) } \\ A^x_1 &{} 0 &{} 0 &{} A^x_4 \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} A^x_1=\sqrt{ \frac{c^2 h p \left( 1+\frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }}, \ A^x_2=\sqrt{\frac{\rho }{k \Gamma }}, \ A^x_3=\sqrt{ \frac{p}{h \Gamma ^5 \rho ^2 \left( 1-{u_x}^2\right) }} \text { and } A^x_4=\sqrt{ \frac{c^2 h p \left( 1- \frac{c {u_x}}{\Gamma \sqrt{Q^x}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }}. \end{aligned}$$

Using Eq. (26), we can calculate the entropy scaled right eigenvectors for the conservative system satisfying (27).

Remark B.1

Proceeding similarly in the y-direction, we get the eigenvalues of the Jacobian matrix as

$$\begin{aligned} \lambda _1^y =\frac{(1-c^2)u_y-(c/\Gamma ) \sqrt{Q^y}}{1-c^2 \varvec{u}^2}, \ \ \lambda _2^y =u_y, \ \ \lambda _3^y =u_y \ \text {and} \ \ \lambda _4^y = \frac{(1-c^2)u_y+(c/\Gamma ) \sqrt{Q^y}}{1-c^2 \varvec{u}^2}, \end{aligned}$$

and the right eigenvector matrix, for the system in primitive variables, as

$$\begin{aligned} \tilde{\mathbf {R}}_w^y =\left( \begin{array}{cccc} \frac{1}{c^2 h} &{} 1 &{} 0 &{} \frac{1}{c^2 h}\\ \frac{\left( c-\Gamma \sqrt{Q^y} {u_y}\right) {u_x}}{c h \Gamma ^2 \rho \left( {u_y}^2-1\right) } &{} 0 &{} 1 &{} \frac{\left( c+\Gamma \sqrt{Q^y} {u_y}\right) {u_x}}{c h \Gamma ^2 \rho \left( {u_y}^2-1 \right) }\\ \frac{-\sqrt{Q^y}}{c h \Gamma \rho } &{} 0 &{} 0 &{} \frac{\sqrt{Q^y}}{c h \Gamma \rho }\\ 1 &{} 0 &{} 0 &{} 1 \end{array} \right) \end{aligned}$$

where $Q^y=1-u_y^2-c^2 u_x^2$. Consequently, the entropy scaled right eigenvector matrix is given by

$$\begin{aligned} {\mathbf {R}}_w^y = \begin{pmatrix} \frac{A^y_1}{c^2 h} &{} A^y_2 &{} 0 &{} \frac{A^y_4}{c^2 h}\\ \frac{A^y_1\left( c-\Gamma \sqrt{Q^y} {u_y}\right) {u_x}}{c h \Gamma ^2 \rho \left( {u_y}^2-1\right) } &{} 0 &{} A^y_3 &{} \frac{A^y_4\left( c+\Gamma \sqrt{Q^y} {u_y}\right) {u_x}}{c h \Gamma ^2 \rho \left( {u_y}^2-1 \right) }\\ \frac{-A^y_1\sqrt{Q^y}}{c h \Gamma \rho } &{} 0 &{} 0 &{} \frac{A^y_4 \sqrt{Q^y}}{c h \Gamma \rho }\\ A^y_1 &{} 0 &{} 0 &{} A^y_4 \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} A^y_1=\sqrt{ \frac{c^2 h \frac{\rho }{\beta } \left( 1+\frac{c {u_y}}{\Gamma \sqrt{Q^y}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }}, \ A^y_2=\sqrt{\frac{\rho }{k \Gamma }}, \ A^y_3=\sqrt{ \frac{\rho }{\beta h \Gamma ^5 \rho ^2 \left( 1-{u_y}^2\right) }}, \ \text {and } A^y_4=\sqrt{ \frac{c^2 h \frac{\rho }{\beta } \left( 1- \frac{c {u_y}}{\Gamma \sqrt{Q^y}}\right) }{2 \Gamma \left( 1-c^2 \varvec{u}^2\right) }}. \end{aligned}$$