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

Christian Klingenberg University of Würzburg Gero Schnücke University of Würzburg Yinhua Xia University of Science and Technology of China

Numerical Analysis and Scientific Computing mathscidoc:1612.25004

Distinguished Paper Award in 2020

Mathematics of Computation, 86, (305), 1203-1232, 2017.5
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, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k+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.
Arbitrary Lagrangian-Eulerian discontinuous Galerkin method, hyperbolic conservation laws, geometric conservation law, cell entropy inequality, error estimates, maximum principle, slope limiter conditions
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@inproceedings{christian2017arbitrary,
  title={Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: analysis and application in one dimension},
  author={Christian Klingenberg, Gero Schnücke, and Yinhua Xia},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161231213553867719700},
  booktitle={Mathematics of Computation},
  volume={86},
  number={305},
  pages={1203-1232},
  year={2017},
}
Christian Klingenberg, Gero Schnücke, and Yinhua Xia. Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: analysis and application in one dimension. 2017. Vol. 86. In Mathematics of Computation. pp.1203-1232. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161231213553867719700.
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