Steger-Warming flux vector splitting method for special relativistic hydrodynamics

Abstract

This paper discusses the properties of the rotational invariance and hyperbolicity in time of the governing equations of the ideal special relativistic hydrodynamics and proves for the first time that the ideal relativistic hydrodynamical equations satisfy the homogeneity property, which is the footstone of the Steger–Warming flux vector splitting method [J. L. Steger and R. F. Warming, J. Comput. Phys., 40(1981), 263–293]. On the basis of this remarkable property, the Steger–Warming flux vector splitting (SW‐FVS) is given. Two high‐resolution SW‐FVS schemes are also given on the basis of the initial reconstructions of the solutions and the fluxes, respectively. Several numerical experiments are conducted to validate the performance of the SW‐FVS method. Copyright © 2013 John Wiley & Sons, Ltd.

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

“…The RHD system (1.1) with (2.1) is identical to the d-dimensional non-relativistic Euler equations in the formal structure, and also satisfies the rotational invariance and the homogeneity as well as the hyperbolicity in time, see [69]. The momentum equations in (1.1) are only with a Lorentz-contracted momentum density replacing ρv i in the nonrelativistic Euler equations.…”

“…After that, various modern shock-capturing methods were gradually developed for the special RHDs since 1990s, e.g. the HLL (Harten-Lax-van Leer-Einfeldt) method [43], the two-shock approximation solvers [1,8], the flux corrected transport method [11], the Roe solver [13], the HLLC (Harten-Lax-van Leer-Contact) approximate Riemann solver [37], the flux-splitting method based on the spectral decomposition [10], Steger-Warming flux vector splitting method [69], and the kinetic schemes [57,27,39]. Besides those, high-order accurate schemes for the RHD system were also studied, e.g.…”

Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

“…Since 1990s, the numerical study of the RHDs began to attract considerable attention, and various modern shock-capturing methods with an exact or approximate Riemann solver have been developed for the RHD equations. Some examples are the local characteristic approach [25], the two-shock approximation solvers [5,8], the Roe solver [13], the flux corrected transport method [12], the flux-splitting method based on the spectral decomposition [11], the piecewise parabolic method [26,33], the HLL (Harten-Lax-van Leer) method [42], the HLLC (Harten-Lax-van Leer-Contact) method [32] and the Steger-Warming flux vector splitting method [59]. The analytical solution of the Riemann problem in relativistic hydrodynamics was studied in [28].…”

Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics

“…Moreover, the adaptive mesh refinement (AMR) techniques were proved to be powerful and useful in simulating RHD, for example [54,43,14] and many software packages have been developed with AMR support and RHD extension, such as the ENZO [27] and RAMSES [41], etc. Additional methods and details can be found in [52,53,46,61,17], and the references therein, as well as the review paper [22].…”