Point Value Multiscale Algorithms for 2D Compressible Flows

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.