03-18
Second Order Accurate Kinetic Schemes for the Ultra-Relativistic Euler Equations
by Kunik, M.; Qamar, S.; Warnecke, G.
Preprint series: 03-18, Preprints
- MSC:
- 65M99 None of the above but in this section
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics, See also {83C55, 85A30}
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 equations, kinetic schemes, second orderaccuracy, conservation laws, hyperbolic systems, entropy conditions, positivity, L1-stability, discontinuous solutions.
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.