Abstract

Since the solution of hyperbolic conservation laws typically exhibits discontinuities, efficient numerical schemes will employ locally refined discretizations that dynamically adapt to the solution. To trigger grid refinement and coarsening an appropriate indicator is needed. Due to the lack of a stable variational formulation for the problem at hand, we apply a multiresolution analysis (MRA) based on multiwavelets to perform data compression. The MRA is combined with a standard discontinuous Galerkin (DG) scheme to end up with an adaptive DG scheme. The framework of both the MRA and the DG scheme will be presented in some detail. The performance of the resulting scheme will be discussed by means of numerous testcases in 1D and 2D for scalar as well as systems of conservation laws. We conclude with a brief outlook how to employ the adaptive framework for the investigation of stochastic conservation laws.