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Journal of Scientific Computing

Erschienen in:

29.05.2017

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

verfasst von: Christian Klingenberg, Gero Schnücke, Yinhua Xia

Erschienen in: Journal of Scientific Computing | Ausgabe 2-3/2017

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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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Titel: An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations
verfasst von: Christian Klingenberg
Gero Schnücke
Yinhua Xia
Publikationsdatum: 29.05.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2-3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0471-2

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