by Kunik, M., Qamar, S., Warnecke, G..

**Series:** 2003-18, Preprints

- MSC:
- 65M99 None of the above, but in this section
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics

**Abstract:**

A second order accurate kinetic scheme for the

numerical solution of the relativistic Euler equations is presented. These

equations describe the flow of a perfect

fluid in terms of the particle density n, the spatial part of the

four-velocity u and the pressure p. The kinetic

scheme, is based on the well-known fact that the relativistic Euler equations

are the moments of the relativistic Boltzmann equation of the kinetic theory of

gases when the distribution function is relativistic Maxwellian.

The kinetic scheme consists of two phases, the convection phase (free-flight) and collision

phase. The velocity distribution function at the end of the free-flight is

the solution of the collisionless transport equation. The collision phase

instantaneously relaxes the distribution to the local Maxwellian distribution. The fluid dynamic

variables of density, velocity, and internal energy are obtained as moments of

the velocity distribution function at the end of the free-flight phase. The scheme presented

here is an explicit method and unconditionally stable. The conservation laws of mass, momentum and energy

as well as the entropy inequality are everywhere exactly satisfied

by the solution of the kinetic scheme. The scheme also satisfies positivity

and $L^1$-stability. The scheme can be easily made into a total variation

diminishing (TVD) method for the distribution function through a suitable choice

of the interpolation strategy. In the numerical case studies the results

obtained from the first- and second-order kinetic schemes are compared with

the first- and second-order upwind and central schemes. We also calculate the

experimental order of convergence (EOC) and numerical $L^1$-stability of the

scheme for the smooth initial data.

**Keywords:**

Relativistic Euler equation