Point Value Multiscale Algorithms for 2D Compressible Flows
Guillaume Chiavassa, Rosa Donat · SIAM Journal on Scientific Computing · 2001
The numerical simulation of physical problems modeled by systems of conservation laws is difficult due to the presence of discontinuities in the solution. High-order shock capturing schemes combine sharp numerical profiles at discontinuities with a highly accurate approximation in smooth regions, but usually their computational cost is quite large. Following the idea of A. Harten [Comm. Pure Appl. Math., 48 (1995), pp. 1305--1342] and Bihari and Harten [SIAM J. Sci. Comput., 18 (1997), pp. 315--354], we present in this paper a method to reduce the execution time of such simulations. It is based on a point value multiresolution transform that is used to detect regions with singularities. In these regions, an expensive high-resolution shock capturing scheme is applied to compute the numerical flux at cell interfaces. In smooth regions a cheap polynomial interpolation is used to deduce the value of the numerical divergence from values previously obtained on lower resolution scales. This method is applied to solve the two-dimensional compressible Euler equations for two classical configurations. The results are analyzed in terms of quality and efficiency.