Multiresolution schemes for conservation laws

In recent years a variety of high{order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive ux evaluations in order to resolve discontinuities accurately. But in large parts of the ow domain the solution is smooth. Hence in these regions an unexpen-sive nite diierence scheme suuces. In order to reduce the number of expensive ux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary tted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of uxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression eeects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two{dimensional Euler equations modeling hy-personic ow around a blunt body.