Dedner, Andreas and Giesselmann, Jan (2016) A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws. SIAM Journal on Numerical Analysis, 54 (6). pp. 3523-3549. doi:10.1137/15M1046265

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Abstract

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines approach combining discontinuous Galerkin spatial discretization with single- or multi-step methods in time. The construction of the estimators requires a reconstruction in time for which we present a very general framework first for odes and then apply the approach to conservation laws. The reconstruction does not depend on the actual method used for evolving the solution in time. Most importantly it covers in addition to implicit methods also the wide range of explicit methods typically used to solve conservation laws. For the spatial discretization, we allow for standard choices of numerical fluxes. We use reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. We study under which conditions on the numerical flux the estimate is of optimal order pre-shock. While the estimator we derive is computable and valid post-shock for fixed meshsize, it will blow up as the meshsize tends to zero. This is due to a breakdown of the relative entropy framework when discontinuities develop. We conclude with some numerical benchmarking to test the robustness of the derived estimator.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics Faculty of Science > Centre for Scientific Computing
Library of Congress Subject Headings (LCSH):	Galerkin methods, Conservation laws (Mathematics) , Differential equations, Hyperbolic
Journal or Publication Title:	SIAM Journal on Numerical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1429
Official Date:	2016
Dates:	Date Event 2016 Published 6 December 2016 Available 29 August 2016 Accepted 2 November 2015 Submitted
Volume:	54
Number:	6
Page Range:	pp. 3523-3549
DOI:	10.1137/15M1046265
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Royal Society (Great Britain), Deutsche Forschungsgemeinschaft (DFG)
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