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An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations

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Abstract

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 \(\mathrm {L}^{\infty }\left( 0,T;\mathrm {L}^{2}\left( \Omega \right) \right) \)-norm. In particular, the optimal (\(k+1\)) convergence in one dimension and the suboptimal (\(k+\frac{1}{2}\)) 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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Authors and Affiliations

  1. University of Würzburg, Emil-Fischer-Str. 40, 97074, Würzburg, Germany

    Christian Klingenberg

  2. University of Cologne, Weyertal 86-90, 50931, Köln, Germany

    Gero Schnücke

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yinhua Xia

Authors
  1. Christian Klingenberg
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  2. Gero Schnücke
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  3. Yinhua Xia
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Corresponding author

Correspondence to Yinhua Xia.

Additional information

Research supported by NSFC Grant Nos. 11471306 and 11371342.

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Klingenberg, C., Schnücke, G. & Xia, Y. An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations. J Sci Comput 73, 906–942 (2017). https://doi.org/10.1007/s10915-017-0471-2

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  • DOI: https://doi.org/10.1007/s10915-017-0471-2

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