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Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

Authors: Christian Klingenberg, Gero Schnücke and Yinhua Xia
Journal: Math. Comp.
MSC (2010): Primary 65M12, 65M15, 65M60; Secondary 35L65
Published electronically: June 20, 2016
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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

Gero Schnücke
Affiliation: University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany

Yinhua Xia
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

Keywords: Arbitrary Lagrangian-Eulerian discontinuous Galerkin method, hyperbolic conservation laws, geometric conservation law, cell entropy inequality, error estimates, maximum principle, slope limiter conditions
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.
Article copyright: © Copyright 2016 American Mathematical Society