Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations
In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solutio...
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| Format: | Report |
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Unspecified
2013
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| _version_ | 1797099290474053632 |
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| author | Rhebergen, S |
| author_facet | Rhebergen, S |
| author_sort | Rhebergen, S |
| collection | OXFORD |
| description | In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the $r$-adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems. |
| first_indexed | 2024-03-07T05:21:41Z |
| format | Report |
| id | oxford-uuid:df22f350-a893-47f1-b8b8-7f6cb5dd4d02 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:21:41Z |
| publishDate | 2013 |
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| spelling | oxford-uuid:df22f350-a893-47f1-b8b8-7f6cb5dd4d022022-03-27T09:37:14ZWell-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equationsReporthttp://purl.org/coar/resource_type/c_93fcuuid:df22f350-a893-47f1-b8b8-7f6cb5dd4d02Mathematical Institute - ePrintsUnspecified2013Rhebergen, SIn this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the $r$-adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems. |
| spellingShingle | Rhebergen, S Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title | Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title_full | Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title_fullStr | Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title_full_unstemmed | Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title_short | Well-balanced $r$-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations |
| title_sort | well balanced r adaptive and moving mesh space time discontinuous galerkin method for the shallow water equations |
| work_keys_str_mv | AT rhebergens wellbalancedradaptiveandmovingmeshspacetimediscontinuousgalerkinmethodfortheshallowwaterequations |