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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
MR Author ID: 605144

Yingda Cheng
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
MR Author ID: 811395

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID: 242268

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