Adaptive multiresolution discontinuous Galerkin schemes for conservation laws (Q2862523)

The concept of multiresolution-based discontinuous Galerkin (MR-DG) schemes is developed, where for the sake of analysis the authors focus on one-dimensional conservation laws. Instead of the standard approach of estimating the error of the solution, an alternative adaptation strategy that does not rely on the existence of some error estimator is proposed. The main idea is to apply data compression by means of a multiresolution analysis and hard thresh holding (cf. [\textit{A. Cohen} et al., Math. Comput. 72, No. 241, 183--225 (2003; Zbl 1010.65035)]). The perturbation error is introduced as the difference of the results obtained by performing computations with the reference scheme and the adaptive scheme (cf. [\textit{B. Cockburn} and \textit{C.-W. Shu}, Math. Comput. 52, No. 186, 411--435 (1989; Zbl 0662.65083)]). In particular, sufficient conditions are derived that ensure a priori estimates of the perturbation error. Numerical results are also presented to demonstrate the efficiency of the proposed concept.