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An arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations

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

Numerical Analysis and Scientific Computing mathscidoc:1903.25005

Journal of Scientific Computing, 73, 906-942, 2017.6
In this paper, an arbitrary Lagrangian–Eulerian local discontinuous Galerkin (ALE-LDG) method for Hamilton–Jacobi equations will be developed, analyzed and numerically tested. This method is based on the time-dependent approximation space defined on the moving mesh. A priori error estimates will be stated with respect to the L∞ 􏰒(0, T ; L2)norm. In particular, the optimal (k + 1) convergence in one dimension and the suboptimal (k + 1 ) convergence in two dimensions will be proven for the semi-discrete method, when a local Lax–Friedrichs flux and piecewise polynomials of degree k on the reference cell are used. Furthermore, the validity of the geometric conservation law will be proven for the fully-discrete method. Also, the link between the piecewise constant ALE-LDG method and the monotone scheme, which converges to the unique viscosity solution, will be shown. The capability of the method will be demonstrated by a variety of one and two dimensional numerical examples with convex and noneconvex Hamiltonian.
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@inproceedings{christian2017an,
  title={An arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations},
  author={Christian Klingenberg, Gero Schnücke, and Yinhua Xia},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190323094402235978222},
  booktitle={Journal of Scientific Computing},
  volume={73},
  pages={906-942},
  year={2017},
}
Christian Klingenberg, Gero Schnücke, and Yinhua Xia. An arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations. 2017. Vol. 73. In Journal of Scientific Computing. pp.906-942. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190323094402235978222.
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