High Order Central Schemes Applied to Relativistic Multi-Component Flow Models

Abstract

The dynamics of inviscid multi-component relativistic fluids may be modeled by the relativistic Euler equations, augmented by one (or more) additional species equation(s). We use the highresolution staggered central schemes to solve these equations. The equilibrium states for each component are coupled in space and time to have a common temperature and velocity. The current schemes can handle strong shocks and the oscillations near the interfaces are negligible, which usually happens in the multi-component flows. The schemes also guarantee the exact mass conservation for each component, the exact conservation of total momentum, and energy in the whole particle system. The central schemes are robust, reliable, compact and easy to implement. Several one-and two-dimensional numerical test cases are included in this paper, which validate the application of these schemes to relativistic multi-component flows. T. Ghaffar et al. 1170 namics have been reported. All these methods are mostly developed out of the existing reliable methods for solving the Euler equations of non-relativistic or Newtonian gas dynamics.The first attempt to solve the equations of relativistic gas dynamics (RGD) was made by Wilson [1] [2] using an Eulerian explicit finite difference code with monotonic transport. The code relies on artificial viscosity technique [3] to handle shock wave. Despite of its popularity it turned out to be unable to accurately describe the extremely relativistic flows [4]. In mid eighties, Norman and Winkler [5] proposed a reformulation of the difference equations with artificial viscosity consistent with relativistic dynamics of non-perfect fluids. Dean et al. [6] used flux correcting algorithms for RGD equations in the context of heavy ion collisions.A good introduction about the recent methods applied to RGD can be found in the review article of Martí and Müller [7]. Some popular methods which are extended for RGD and are also discussed in [7] are the Rao methods [8] used by Eulderink et al. [9] [10], PPM method [11] by Martí and Müller [12], Glimm's methods [13] by Wen et al. [14], HLL method [15] by Schneider et al. [16], Marquina flux formula [17] by Martí et al. [12] [18] and relativistic beam scheme [19] by Yang et al. [20]. The development of numerical methods for the non-relativistic multi-component flows have attracted much attention in the past years, for example Fedkiw et al. [21] [22], Karni [23]-[25], Karni and Quirk [26], Marquina and Mulet [27]. Moreover, Xu [28] used BGK-based gas-kinetic schemes to solve multi-component flows, while Lian and Xu [29] used the same scheme in order to solve the multi-component flows with chemical reactions.This paper is an extension of the relativistic Euler equations to multi-component flows. We use the high-resolution non-oscillatory central schemes of Nessyahu and Tadmor [30] as well as Jiang and Tadmor [31] to solve these Euler equations. In this study, we consider only two-component flow, however, an extension to further components will result in addition of a ...