A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations

A gas-kinetic solver is developed for the ultrarelativistic Euler equations. The scheme is based on the direct splitting of the flux function of the Euler equations with inclusion of ``particle'' collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the usual kinetic flux-vector splitting (KFVS) schemes which are based on the free particle transport at the cell interfaces in the gas evolution stage. Although in a usual KFVS scheme the free particle transport gives robust solutions, it gives a smeared solution at the contact discontinuities. The new BGK-type KFVS scheme solves this problem and gives robust and reliable solutions as well as good resolution at the contact discontinuity. In contrast to the classical kinetic theory, we have only a finite domain of dependence due to the structure of the light cone. The scheme is naturally multidimensional and is extended to the two-dimensional case in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction. In the numerical case studies the results obtained from the BGK-type KFVS schemes are compared with the exact Riemann solution, KFVS schemes, upwind schemes, and central schemes.

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