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Convergence of discontinuous Galerkin schemes for front propagation with obstacles

Authors: Olivier Bokanowski, Yingda Cheng and Chi-Wang Shu
Journal: Math. Comp. 85 (2016), 2131-2159
MSC (2010): Primary 65-XX; Secondary 65M60, 65M12
Published electronically: December 29, 2015
MathSciNet review: 3511277
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Abstract: We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $ \min (u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These ``low regularity'' assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.

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Additional Information

Olivier Bokanowski
Affiliation: Université Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France

Yingda Cheng
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Keywords: Hamilton-Jacobi-Bellman equations, discontinuous Galerkin methods, level sets, front propagation, obstacle problems, dynamic programming principle, convergence
Received by editor(s): June 13, 2013
Received by editor(s) in revised form: July 4, 2014, and February 21, 2015
Published electronically: December 29, 2015
Additional Notes: The research of the first author was supported by the EU under the 7th Framework Programme Marie Curie Initial Training Network “FP7-PEOPLE-2010-ITN”, SADCO project, GA number 264735-SADCO.
The research of the second author was supported by NSF grant DMS-1217563 and the start-up grant from Michigan State University.
The research of the third author was supported by ARO grant W911NF-11-1-0091 and NSF grants DMS-1112700 and DMS-1418750.
Article copyright: © Copyright 2015 American Mathematical Society

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