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.


Preprint series: 06-20, Preprints

35-04 Explicit machine computation and programs (not the theory of computation or programming)
35L60 Nonlinear first-order PDE of hyperbolic type
35L65 Conservation laws
35L67 Shocks and singularities, See also {58C27, 76L05}
35Q05 Euler-Poisson-Darboux equation and generalizations
35Q75 PDE in relativity


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

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