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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension
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by Christian Klingenberg, Gero Schnücke and Yinhua Xia PDF
Math. Comp. 86 (2017), 1203-1232 Request permission

Abstract:

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, $\mathrm {L}^{2}$ stability and error estimates are proven. More precisely, we prove the sub-optimal ($k+\frac {1}{2}$) convergence for monotone fluxes and optimal ($k+1$) convergence for an upwind flux when a piecewise $P^k$ polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover, we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.
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Additional Information
  • Christian Klingenberg
  • Affiliation: University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
  • MR Author ID: 221691
  • Email: klingen@mathematik.uni-wuerzburg.de
  • Gero Schnücke
  • Affiliation: University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
  • Email: gero.schnuecke@web.de
  • Yinhua Xia
  • Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
  • MR Author ID: 819188
  • Email: yhxia@ustc.edu.cn
  • Received by editor(s): June 5, 2015
  • Received by editor(s) in revised form: September 25, 2015
  • Published electronically: June 20, 2016
  • Additional Notes: The third author is the corresponding author. The research of the third author was supported by NSFC grants No. 11371342 and No. 11471306.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1203-1232
  • MSC (2010): Primary 65M12, 65M15, 65M60; Secondary 35L65
  • DOI: https://doi.org/10.1090/mcom/3126
  • MathSciNet review: 3614016