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Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

Authors: Pei Fu, Gero Schnücke and Yinhua Xia
Journal: Math. Comp. 88 (2019), 2221-2255
MSC (2010): Primary 35L65, 65M60
Published electronically: March 5, 2019
MathSciNet review: 3957892
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Abstract: In Klingenberg, Schnücke, and Xia (Math. Comp. 86 (2017), 1203–1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semidiscrete method the $\mathrm {L}^2$-stability will be proven. Furthermore, an error estimate which provides the suboptimal ($k+\frac {1}{2}$) convergence with respect to the $\mathrm {L}^{\infty }\!\left (0,T;\mathrm {L}^{2}\!\left (\Omega \right )\right )$-norm will be presented when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree $k$. The two-dimensional fully-discrete explicit method will be combined with the bound-preserving limiter developed by Zhang, Xia, and Shu (in J. Sci. Comput. 50 (2012), 29–62). This limiter does not affect the high-order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter, the validity of a discrete maximum principle will be proven. The numerical stability, robustness, and accuracy of the method will be shown by a variety of two-dimensional computational experiments on moving triangular meshes.

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Additional Information

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

Gero Schnücke
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Köln, Germany

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

Keywords: Arbitrary Lagrangian-Eulerian discontinuous Galerkin method, conservation laws, moving simplex meshes, geometric conservation law, $\mathrm {L}^2$-stability, error estimates, maximum principle
Received by editor(s): April 10, 2018
Received by editor(s) in revised form: September 29, 2018
Published electronically: March 5, 2019
Additional Notes: The third author’s research was supported by NSFC grants No. 11871449 and No. 11471306, and a grant from the Science & Technology on Reliability & Environmental Engineering Laboratory (No. 6142A0502020817).
The third author is the corresponding author.
Article copyright: © Copyright 2019 American Mathematical Society