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Volume 30, Issue 2
Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations

Qian Zhang, Yan Xu & Chi-Wang Shu

DOI: 10.4208/cicp.OA-2020-0027

Commun. Comput. Phys., 30 (2021), pp. 321-356.

Published online: 2021-05

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  • Abstract

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly.

  • Keywords

Discontinuous Galerkin method, Fornberg-Whitham type equation, dissipative scheme, conservative scheme, error estimates.

  • AMS Subject Headings

65M60, 35L75, 35G25

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{CiCP-30-321, author = {Zhang , QianXu , Yan and Shu , Chi-Wang}, title = {Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {2}, pages = {321--356}, abstract = {

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0027}, url = {http://global-sci.org/intro/article_detail/cicp/19117.html} }
TY - JOUR T1 - Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations AU - Zhang , Qian AU - Xu , Yan AU - Shu , Chi-Wang JO - Communications in Computational Physics VL - 2 SP - 321 EP - 356 PY - 2021 DA - 2021/05 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0027 UR - https://global-sci.org/intro/article_detail/cicp/19117.html KW - Discontinuous Galerkin method, Fornberg-Whitham type equation, dissipative scheme, conservative scheme, error estimates. AB -

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly.

Qian Zhang, Yan Xu & Chi-Wang Shu. (2021). Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations. Communications in Computational Physics. 30 (2). 321-356. doi:10.4208/cicp.OA-2020-0027
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