Adaptive multiresolution discontinuous Galerkin schemes for conservation laws

A multiresolution-based adaptation concept is proposed that aims at accelerating discontinuous Galerkin schemes applied to non-linear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multireso...

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Bibliographic Details
Published in:Mathematics of computation 2014-01, Vol.83 (285), p.113-151
Main Authors: HOVHANNISYAN, NUNE, MÜLLER, SIEGFRIED, SCHÄFER, ROLAND
Format: Article
Language:English
Subjects:
Approximation
Coefficients
Conservation laws
Entropy
Error rates
Evolution equations
Index sets
Mathematical discontinuity
Scalars
Threshing
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Summary:A multiresolution-based adaptation concept is proposed that aims at accelerating discontinuous Galerkin schemes applied to non-linear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multiresolution analysis is performed 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. A central mathematical problem addressed in this work is then to show at least for scalar one-dimensional problems that choosing an appropriate threshold value, the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh. Numerical comparisons demonstrate the efficiency of the concept.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-2013-02732-9