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A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws
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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 ISSN 0036-1429.
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Official URL: http://dx.doi.org/10.1137/15M1046265
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 | ||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Centre for Scientific Computing |
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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: |
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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 | ||||||||||
Date of first compliant deposit: | 4 October 2016 | ||||||||||
Date of first compliant Open Access: | 15 December 2016 | ||||||||||
Funder: | Royal Society (Great Britain), Deutsche Forschungsgemeinschaft (DFG) | ||||||||||
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