1 Introduction
Magnetohydrodynamics (MHDs) describe the dynamics of electricallyconducting fluids in the presence of magnetic field and play an important role in many fields including astrophysics, plasma physics and space physics. In many cases, astrophysics and high energy physics often involve fluid flow at nearly speed of light so that the relativistic effect should be taken into account. Relativistic MHDs (RMHDs) have applications in investigating astrophysical scenarios from stellar to galactic scales, e.g., gammaray bursts, astrophysical jets, core collapse supernovae, formation of black holes, and merging of compact binaries.
In the dimensional space, the governing equations of special RMHDs can be written as a system of hyperbolic conservation laws
(1.1) 
along with an additional divergencefree condition on the magnetic field
(1.2) 
where or . Here we employ the geometrized unit system so that the speed of light in vacuum is equal to one. In (1.1
), the conservative vector
, and the flux in the direction is defined bywith the mass density , the momentum density vector , the energy density , and the vector denoting the th column of the unit matrix of size . Additionally, is the restmass density, denotes the fluid velocity vector, is the Lorentz factor, is the total pressure containing the gas pressure and magnetic pressure , represents the specific enthalpy, and is the specific internal energy. Different from the nonrelativistic case, the variables and depend on the magnetic field nonlinearly. The system (1.1) is closed with an equation of state. In this paper, we consider the ideal equation of state , where is constant and denotes the adiabatic index.
The system (1.1) involves strong nonlinearity, making its analytic treatment quite difficult. Numerical simulation is a primary approach to explore physical laws in RMHDs. Since nearly 2000s, numerical study of RMHD has attracted considerable attention, and various numerical methods have been developed for the RMHD equations, for example, the total variation diminishing scheme [2], adaptive mesh methods [49, 31], discontinuous Galerkin methods [57, 58], entropy limited appraoch [28] and the highorder schemes based on subluminal reconstruction [4], etc. Recently, physicalconstraintspreserving schemes were proposed for relativistic hydrodynamics [55, 51] and RMHDs [56], A systematic review of numerical RMHD schemes can be found in [39]. Besides the standard difficulty in solving the nonlinear hyperbolic systems, an additional numerical challenge for the RMHD system (1.1) comes from the divergencefree condition (1.2), which is also involved in the nonrelativistic ideal MHD system. Numerical preservation of (1.2) is highly nontrivial (for ) but crucial for the robustness of numerical computation. Numerical experiments and analysis in the nonrelativistic MHD case indicated that violating the divergencefree condition (1.2) may lead to numerical instability and nonphysical solutions [9, 3, 52]. Various numerical techniques were proposed to reduce or control the effect of divergence error; see, e.g., [19, 44, 48, 15, 3, 36, 37, 58, 53, 54].
Due to the nonlinear hyperbolic nature of the RMHD equations (1.1), solutions of (1.1) can be discontinuous with the presence of shocks or contact discontinuities. This leads to consideration of weak solutions. However, the weak solutions may not be unique. To select the “physically relevant” solution among all weak solutions, entropy conditions are usually imposed as the admissibility criterion. In the case of RMHD equations (1.1), there is a natural entropy condition arising from the second law of thermodynamics which should be respected. It is natural to seek entropy stable numerical schemes which satisfy a discrete version of entropy condition. Entropy stable numerical methods ensure that the entropy is conserved in smooth regions and dissipated across discontinuities. Thus, the numerics precisely follow the physics of the second law of thermodynamics and can be more robust. Moreover, entropy stable schemes also allow one to limit the amount of dissipation added to the schemes to guarantee the entropy stability. For the above reasons, developing entropy stable schemes for RMHD equations (1.1) is meaningful and highly desirable.
Entropy stability analysis was well studied for firstorder accurate schemes and scalar conservation laws with all entropy functions in early work [14, 30, 42, 43]. In recent years, there has been much interest in exploring highorder accurate entropy stable schemes for systems of hyperbolic conservation laws, with entropy stability focused on single given entropy function. Tadmor [46, 47] established the framework of entropy conservative fluxes, which conserves entropy locally, and entropy stable fluxes for secondorder schemes. Lefloch, Mercier and Rohde [35] proposed a produce to construct higherorder accurate entropy conservative fluxes. Fjordholm, Mishra and Tadmor [21] developed a methodology for constructing highorder accurate entropy stable schemes, which combine highorder entropy conservative fluxes and suitable numerical dissipation operators based on essentially nonoscillatory (ENO) reconstruction that satisfies the sign property [22]. Highorder entropy stable schemes have also been constructed via the summationbyparts procedure [20, 10, 24]. Entropy stable space–time discontinuous Galerkin (DG) schemes were studied in [5, 6, 33], where the exact integration is required for the proof of entropy stability. More recently, a unified framework was proposed in [13] for designing provably entropy stable highorder DG methods through suitable numerical quadrature. There are other studies that address various aspects of entropy stability, including but not limited to [8, 23, 7, 32]. As a key ingredient in designing entropy stable schemes, the construction of affordable entropy conservative fluxes has received much attention. Although there is a general way to construct entropy conservative flux based on path integration [46, 47], the resulting flux may not have an explicit formula and can be computationally expensive. Explicit entropy conservative fluxes were derived for the Euler equations [34, 11, 45], shallow water equations [25], special relativistic hydrodynamics without magnetic field [18], and ideal nonrelativistic MHD equations [12, 50], etc. Different from the Euler equations, the conservative form of ideal nonrelativistic MHD equations is not symmetrizable and does not admit an entropy [27, 5, 12]. Entropy symmetrization can be achieved by a modified nonrelativistic MHD system with an additional source [27, 5]. Based on the modified formulations, several entropy stable schemes were well developed for nonrelativistic MHDs; see, e.g., [12, 16, 50, 38, 17].
This paper aims to present entropy analysis and highorder accurate entropy stable finite difference schemes for RMHD equations on Cartesian meshes. In comparison with the nonrelativistic MHD case, the difficulties in this work mainly come from (i) the unclear symmetrizable form of RMHD equations that admits an entropy pair, and (ii) the highly nonlinear coupling between RMHD equations (1.1), which leads to no explicit expression of the primitive variables , entropy variables and the flux in terms of . The effort and findings in this work include the following:

We show that the conservative form (1.1) of RMHD equations is not symmetrizable and thus does not admit an entropy pair. A modified RMHD system is proposed by building the divergencefree condition (1.2) into the equations through adding a source term, which is proportional to . The modified RMHD system is symmetrizable, possesses the natural entropy pair, and can be considered as the relativistic extension of the nonrelativistic symmetrizable MHD system proposed by Godunov [27]. As a result, it has almost all good features that the nonrelativistic symmetrizable MHD system possesses.

We derive a consistent twopoint entropy conservative numerical flux for the symmetrizable RMHD equations. The flux has an explicit analytical formula, is easy to compute and thus is affordable. The key is to carefully select a set of parameter variables which can explicitly express the entropy variables and potential fluxes in simple form. Due to the presence of the source term in the symmetrizable RMHD system, the standard framework [47] of the entropy conservative flux is not applicable here and should be modified to take the effect of the source term into account.

We construct semidiscrete highorder accurate entropy conservative schemes and entropy stable schemes for symmetrizable RMHD equations on Cartesian meshes. The secondorder entropy conservative schemes are built on the proposed twopoint entropy conservative flux and a central difference type discretization to the source term. Higherorder entropy conservative schemes are constructed by suitable linear combinations [35] of the proposed twopoint entropy conservative flux. We find that, to ensure the entropy conservative property and highorder accuracy, a particular discretization of the symmetrization source term should be employed, which becomes a key ingredient in these highorder schemes. Arbitrarily highorder accurate entropy stable schemes are obtained by adding suitable dissipation terms into the entropy conservative schemes.
The paper is organized as follows. After giving the entropy analysis in Sect. 2, we derive the explicit twopoint entropy conservative fluxes in Sect. 3. Onedimensional (1D) entropy conservative schemes and entropy stable schemes are constructed in Sect. 4, and their extensions to two dimensions (2D) are presented in Sect. 5. We conduct numerical tests in Sect. 6 to verify the performance of the proposed highorder accurate entropy stable schemes, before concluding the paper in Sect. 7.
2 Entropy Analysis
First, let us recall the definition of entropy function.
Definition 2.1.
A convex function is called an entropy for the system (1.1) if there exist entropy fluxes such that
(2.1) 
where the gradients and are written as row vectors, and is the Jacobian matrix. The functions form an entropy pair.
If a hyperbolic system of conservation laws admits an entropy pair, then the smooth solutions of the system should satisfy
For nonsmooth solutions, the above identity does not hold in general and is replaced with an inequality
(2.2) 
which is interpreted in the week sense and known as the entropy condition.
The existence of an entropy pair is closely related to the symmetrization of a hyperbolic system of conservation laws [26].
Definition 2.2.
Lemma 2.1.
2.1 Entropy Function for RMHD Equations
It is natural to ask whether the RMHD equations (1.1) admit an entropy pair or not.
Let us consider the thermodynamic entropy For smooth solutions, the RMHD equations (1.1) can be used to derive an equation for :
(2.4) 
Then under the divergencefree condition (1.2), the following quantities
(2.5) 
satisfy an additional conservation law for smooth solution, thus may be an entropy function.
Theorem 2.1.
The entropy variables corresponding to the entropy function are given by
(2.6) 
Proof.
Since cannot be explicitly expressed by , direct derivation of can be quite difficult. Here we consider the following primitive variables
(2.7) 
As and can be explicitly formulated in terms of , it is easy to derive that
and
(2.8) 
where , and and
denote the zero square matrix and the identity matrix of size
, respectively. One can verify that the vector satisfies which implies The proof is complete.However, the change of variable fails to symmetrize the RMHD equations (1.1), and the functions defined in (2.5) do not form an entropy pair for the RMHD equations (1.1); see the following theorem.
Theorem 2.2.
Proof.
We only prove the conclusions for , as the proofs for are similar.
Let us first show that is not symmetric in general. Because cannot be formulated explicitly in terms of , we calculate the Jacobian matrix with the aid of primitive variables . The Jacobian matrix is computed as
(2.10) 
with , , and
Since can be explicitly expressed in terms of , one can derive that
(2.11) 
with Then, we obtain by the inverse of the matrix , i.e.
(2.12) 
with
By the chain rule
, we get the expression of and find it is not symmetric in general. For example, the element of the Jacobian matrix iswhile the element is
2.2 Modified RMHD Equations and Entropy Symmetrization
To address the above issue, we propose a modified RMHD system
(2.13) 
where
(2.14) 
The system (2.13) can be considered as the relativistic extension of the Godunov–Powell system [27, 44] in the ideal nonrelativistic MHDs. The righthand side term of (2.13) is proportional to . This means, at the continuous level, the modified form (2.13) and conservative form (1.1) are equivalent under the condition (1.2). However, the “source term” modifies the character of the RMHD equations, making the system (2.13) symmetrizable and admit the convex entropy pair , as shown below.
Lemma 2.2.
Let In terms of the entropy variables in (2.6), is a homogeneous function of degree one, i.e.,
(2.15) 
In addition, the gradient of with respect to equals , i.e.,
(2.16) 
Proof.
Theorem 2.3.
Proof.
Define
(2.17)  
(2.18) 
which satisfy In terms of the variables , the gradients of and satisfy the following identities
(2.19) 
Substituting (2.19) into (2.13), we can rewrite the modified RMHD equations as
(2.20) 
where the Hessian matrices , and are all symmetric. Moreover, is a convex function on and the matrix is positive definite. Hence, the change of variables symmetrizes the modified RMHD equations (2.13). According to Lemma 2.1, the system (2.13) possesses a strictly convex entropy .
Remark 2.1.
We note that, in the modified RMHD system (2.13), the induction equation is given by Taking the divergence of this equation gives
Combining the continuity equation of (2.13), it yields
(2.21) 
Equation (2.21) implies that the quantity is constant along particle paths. Similar property holds for the Godunov–Powell modified system in the nonrelativistic ideal MHDs [27, 44]. Following Powell’s perspective for the nonrelativistic ideal MHDs [44], it reasonable to conjecture that suitable discretization of the modified RMHD system (2.13) can provide more stable numerical simulation as the numerical divergence error may be advected away by the flow.
3 Derivation of Twopoint Entropy Conservative Fluxes
In this section, we derive explicit twopoint entropy conservative numerical fluxes, which will play an important role in constructing entropy conservative schemes and entropy stable schemes, for the RMHD equations (2.13). Similar to the nonrelativistic MHD case [12], the standard definition [47] of the entropy conservative flux is not applicable here and should be slightly modified, due to the presence of the term in the symmetrizable RMHD equations (2.13). Here we adopt a definition similar to the one proposed in [12] for the nonrelativistic MHD equations.
Definition 3.1.
For a consistent twopoint numerical flux is entropy conservative if
(3.1) 
where , , and are the entropy variables defined in (2.6), the function defined in Lemma 2.2, the potential fluxes defined in (2.18), and the magnetic field component , respectively. The subscripts and indicate that those quantities are corresponding to the “left” state and the “right” state , respectively.
Now, we would like to construct explicit entropy conservative fluxes satisfying the condition (3.1). For notational convenience, we employ
to denote, respectively, the jump and the arithmetic mean of a quantity. In addition, we also need the logarithmic mean
which was first introduced in [34]. Then, one has following identities
(3.2)  
(3.3)  
(3.4) 
which will be frequently used in the following derivation.
Let us introduce the following set of variables
(3.5) 
with and . Define
An explicit twopoint entropy conservative flux for is given below.
Theorem 3.1.
Proof.
Let and . Then the condition (3.1) for can be rewritten as
(3.7) 
To determine the unknown components of , we would like to expand each jump term in (3.7) into linear combination of the jumps of certain parameter variables. This will give us a linear algebraic system of eight equations for the unknown components . There are many options to choose different sets of parameter variables, which may result in different fluxes. Here we take as the parameter variables, to make the resulting formulation of simple.
In terms of the parameter variables , the entropy variables in (2.6) can be explicitly expressed as
and and can be expressed as
Then, using the identities (3.2)–(3.4), we rewrite the jump terms involved in (3.7) as
with
(3.8)  
(3.9) 
Substituting the above expressions of jumps into (3.7) gives
(3.10) 
where
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