by Kunik, M., Shamsul, Q., Warnecke, G..

**Series:** 2003-04, Preprints

- MSC:
- 65M99 None of the above, but in this section
- 65Y20 Complexity and performance of numerical algorithms

**Abstract:**

A gas-kinetic solver is developed for the ultra-relativistic 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

artifical 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 solution, it gives 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. The scheme is naturally multidimensional and is extended to the two-

dimensional case in a usual dimensionally split manner, that is, the formulaefor the

fluxes can be used along each coordinate direction. The high-order resolution of the

scheme is achieved by using MUSCL-type initial reconstruction. In the numerical

case studies the results obtained from the BGK-type KFVS schemes are compared

with the exact solution, KFVS schemes, upwind schemes and central schemes.

**Keywords:**

hyperbolic systems, ultra-relativistic Euler equations, BGK-type KFVS scheme, numerical dissipation, higher order accuracy