|Hyperbolic systems non-relativistic and relativistic Euler equations kinetic schemes conservation laws discontinuous solutions high order accuracy|
Kinetic schemes for the relativistic Euler equations are presented, which describe the flow of a perfect gas in terms of the particle density, the spatial part of the four-velocity and the pressure. The physical frame in the whole study will be exclusively special relativity. We consider both, the ultra-relativistic Euler equations, and a more general form of the relativistic Euler equations. The general form of relativistic Euler equations covers the whole range from the non-relativistic to the ultra-relativistic limit. We also consider as a special case the non-relativistic theory. The basic ingredients of the kinetic schemes are the phase density in equilibrium and the free-flight. The phase density generalizes the non-relativistic Maxwellian for a gas in local equilibrium. The free-flight is given by solutions of a collision free kinetic transport equation. The kinetic schemes presented here are discrete in time but continuous in space. The schemes are explicit and unconditionally stable, i.e., no Courant-Friedrichs-Levy (CFL) condition is needed. Also the schemes are truly multi-dimensional as they cover all the directions of wave propagation in the gas evolution stage. These kinetic schemes preserve the positivity of particle density and pressure for all times and hence are L1-stable. The schemes satisfy the weak form of conservation laws for mass, momentum, and energy, as well as an entropy inequality in any arbitrary domain. The schemes also satisfy the total variation diminishing (TVD) property for the distribution function through a suitable choice of the interpolation strategy. We also extend the schemes to account for the boundary conditions. The kinetic schemes described above are first order in time and space. We also extend the schemes to second order for the one- and two-dimensional ultra-relativistic Euler equations.
In addition, we develop another type of kinetic schemes for the ultra-relativistic Euler equations which are discrete both in time and space. These are an upwind conservative form of the kinetic schemes in which the fluxes are the moments of the relativistic free-flight phase density. We use flux vector splitting in order to calculate the free-flight moment integrals under a natural CFL condition due to the structure of light cone, since every signal speed is bounded by the velocity of light. The schemes are then called kinetic flux vector splitting (KFVS) schemes. Since KFVS schemes are based on the free particle transport at the cells interface in the gas evolution stage, they give smeared solutions especially at the contact discontinuity. To overcome this problem ``particle'' collisions are included in the transport process. Consequently, the artificial dissipation in the schemes are much reduced in comparison with the usual KFVS schemes. These new upwind schemes are called BGK-type KFVS schemes. For the ultra-relativistic Euler equations we have to evaluate the free-flight moment integrals over the compact unit sphere due to the finite domain of dependence in the relativistic kinetic theory. But in the classical kinetic schemes the free-flight moment integrals have infinite integration limits, therefore they need some error-functions which have to be cutoff at their tails. Our schemes are extended to the two-dimensional case in a usual dimensionally split manner. We use a MUSCL-type initial reconstruction for the second order accuracy.
For the comparison of the numerical results, we give the results of exact Riemann solver and Godunov scheme for the one-dimensional ultra-relativistic Euler equations. We also present the central schemes and apply them to both non-relativistic and relativistic Euler equations. The main advantages of the central schemes are compactness and simplicity. We have carried out several one- and two-dimensional numerical test case computations. It was found that kinetic schemes have a comparable accuracy with the upwind and central schemes.