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A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

Izem, Nouh; Seaid, Mohammed

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography Thumbnail


Authors

Nouh Izem



Abstract

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-LobattoLegendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

Citation

Izem, N., & Seaid, M. (2022). A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography. Advances in applied mathematics and mechanics, 14(3), 725-758. https://doi.org/10.4208/aamm.oa-2020-0364

Journal Article Type Article
Acceptance Date May 11, 2021
Publication Date 2022
Deposit Date Nov 2, 2021
Publicly Available Date Nov 2, 2021
Journal Advances in Applied Mathematics and Mechanics
Print ISSN 2070-0733
Electronic ISSN 2075-1354
Publisher Global Science Press
Peer Reviewed Peer Reviewed
Volume 14
Issue 3
Pages 725-758
DOI https://doi.org/10.4208/aamm.oa-2020-0364
Public URL https://durham-repository.worktribe.com/output/1225941
Publisher URL https://www.global-sci.org/aamm

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Copyright Statement
First published in Advances in Applied Mathematics and Mechanics in 14, no.3 (2022), published by Global Science Press.





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