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2003-18

Second Order Accurate Kinetic Schemes for the Ultra-Relativistic Euler Equations

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