Year: 2020
Author: Junming Duan, Huazhong Tang
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 1 : pp. 58–83
Abstract
This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulae of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1902-m2018-0189
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 1 : pp. 58–83
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Hamilton-Jacobi equation ADER Discontinuous Galerkin methods Local continuous spacetime Galerkin predictor High order accuracy.
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