On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state

Abstract

The paper studies the physical-constraints-preserving (PCP) schemes for multidimensional special relativistic magnetohydrodynamics with a general equation of state (EOS) on more general meshes. It is an extension of the work (Ref. [45]) which focuses on the ideal EOS and uniform Cartesian meshes. The general EOS without a special expression poses some additional difficulties in discussing the mathematical properties of admissible state set with the physical constraints on the fluid velocity, density and pressure. Rigorous analyses are provided for the PCP property of finite volume or discontinuous Galerkin schemes with the Lax-Friedrichs (LxF) type flux on a general mesh with non-self-intersecting polytopes. Those are built on a more general form of generalized LxF splitting property and a different convex decomposition technique. It is shown in theory that the PCP property is closely connected with a discrete divergence-free condition, which is proposed on the general mesh and milder than that in Ref. [45].where the conservative vector U = D, m, B, E ⊤ and the flux in the x i -direction F i (U) is defined byCorresponding author. Tel: +86-10-62757018; Fax: +86-10-62751801.where the adiabatic index Γ ∈ (1, 2]. The system (1) takes into account the relativistic description for the dynamics of electrically-conducting fluid (plasma) at nearly speed of light in vacuum in the presence of magnetic fields. The relativistic magneto-fluid flow appears in investigating numerous astrophysical phenomena from stellar to galactic scales, e.g., core collapse super-novae, coalescing neutron stars, Xray binaries, active galactic nuclei, formation of black holes, super-luminal jets and gamma-ray bursts etc. However, due to the relativistic effect, especially the appearance of Lorentz factor, the system (1) involves strong nonlinearity, making its analytic treatment extremely difficult. Numerical simulation is a primary and powerful approach to improve our understanding of the physical mechanisms in the RMHDs. In comparison with the non-relativistic MHD case, the numerical difficulties mainly come from highly nonlinear coupling between the RMHD equations in (1), which leads to no explicit expression of the primitive variables (ρ, v, p) and the flux F i in terms of U.Since nearly 2000s, numerical study of the RMHDs has attracted considerable attention, and various modern shock-capturing methods have been developed for the RMHD equations. They include but are not limited to: the Godunov-type scheme based on the linear Riemann solver [22], the total variation diminishing scheme [2], the third-order accurate central-type scheme based on two-speed approximate Riemann solver [12], the high-order kinetic flux-splitting method [33], the exact Riemann solver [16], the HLLC (Harten-Lax-van Leer-contact) type schemes [20,21,29], the adaptive methods with mesh refinement [1,40], the adaptive moving mesh method [18], the locally divergence-free Runge-Kutta discontinuous Galerkin (RKDG) method and exactly divergence-free central RKDG method with ...

Numerical relativity in spherical coordinates: A new dynamical spacetime and general relativistic MHD evolution framework for the Einstein Toolkit

“…In future versions of the code, problems related to the primitive recovery could be handled by more modern algorithms, such as evolving the entropy S and using it to recover the pressure [42], or using different primitive recovery schemes, see [153] for an overview. Another attractive approach could be the use of physical-constraintpreserving methods [151,154,155].…”

High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

“…The readers are referred to the early review articles [15,27,28] and the references therein. Recently, the properties of the admissible state set and the physical-constraints-preserving (PCP) numerical schemes were well studied for the RHD, see [41,46,48] and [35], and for the special relativistic magnetohydrodynamics [45,47]. The PCP schemes satisfy that both the rest-mass density and the kinetic pressure are positive and the magnitude of the fluid velocity is less than the speed of light.…”

“…The PCP schemes satisfy that both the rest-mass density and the kinetic pressure are positive and the magnitude of the fluid velocity is less than the speed of light. Motivated by [45,47], the positivity-preserving schemes for the non-relativistic ideal magnetohydrodynamics were successfully studied in [42,43]. It is well known that the weak solution of the quasi-linear hyperbolic conservation laws might not be unique so that the entropy conditions are needed to single out the physical relevant solution among all weak solutions.…”

Second-order accurate BGK schemes for the special relativistic hydrodynamics with the Synge equation of state

“…Recently some physical-constraints-preserving (PCP) schemes were developed for the special RHDs and relativistic magnetohydrodynamics (RMHD). They are the high-order accurate PCP finite difference WENO schemes and discontinuous Galerkin (DG) methods proposed in [51,53,41,26,52,54]. The entropystable schemes were also developed for the special RHD or RMHD equations [11,10,9].…”