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2006-20

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

by Heineken, W., Kunik, M..


Series: 2006-20, Preprints

MSC:
35-04 Explicit machine computation and programs (not the theory of computation or programming)
35L60 Nonlinear first-order hyperbolic equations
35L65 Conservation laws
35L67 Shocks and singularities
35Q05 ~Euler-Poisson-Darboux equations
35Q75 PDEs in connection with relativity and gravitational theory

Abstract:
A finite volume method with grid adaption is applied
to two hyperbolic problems: the ultra-relativistic Euler equations,
and a scalar conservation law. Both problems are considered
in two space dimensions and share the common feature
of moving shock waves. In contrast to the classical Euler equations,
the derivation of appropriate initial conditions
for the ultra-relativistic Euler equations is a non-trivial problem
that is solved using one-dimensional shock conditions
and the Lorentz invariance of the system.
The discretization of both problems is based on a finite volume method
of second order in both space and time on a triangular grid.
We introduce a variant of the min-mod limiter that avoids unphysical
states for the Euler system. The grid is adapted during the integration
process. The frequency of grid adaption is controlled automatically
in order to guarantee a fine resolution of the moving shock fronts.
We introduce the concept of ``width refinement'' which enlarges the
width of strongly refined regions around the shock fronts;
the optimal width is found by a numerical study.
As a result we are able to improve efficiency by decreasing
the number of adaption steps. The performance of the finite volume
scheme is compared with several lower order methods.

Keywords:
Scalar conservation laws, ultra-relativistic Euler equations, Riemann problem, shock waves, Lorentz-transformations, finite volume method, min-mod limiter, grid adaption