Adaptive Multiresolution Discontinuous Galerkin Schemes for Conservation Laws: Multi–Dimensional Case

The concept of multiresolution-based adaptive DG schemes for nonlinear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in N. Hovhannisyan, S. Müller, R. Schäfer, Adaptive multiresolution Discontinuous Galerkin Schemes for Conservation Laws, Math. Comp., 2013. The key idea is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and nonlinear scalar conservation laws.