Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

Multiresolution Schemes for Conservation Laws

In recent years a large variety of high order schemes for the simulation of flows were developed. Neglecting dissipative effects these problems are described by hyperbolic conservation laws. Their physical meaningful solution can be approximated by finite-volume methods in conservative form. These schemes involve complex flux evaluations in order to resolve shocks and contact discontinuities. However, such discontinuities only occur on lower dimensional manifolds of the flow field. Only there it is necessary to employ the expensive scheme while elsewhere a simple finite-difference scheme is sufficient. In order to reduce the number of flux evaluations in these domains as much as possible, A. Harten developed a general multiresolution strategy for the one-dimensional case. This results in a new method which is both adaptive and hybrid. It replaces the expensive flux evaluation by interpolation of surrounding fluxes in domains where the solution is smooth. Harten's concept can be realized by using biorthogonal wavelets due to A. Cohen, I. Daubechies and J.-C. Feauveau. The generalization to multi-dimensional problems was also possible. Finally, some numerical results for the two-dimensional Euler equations are presented.