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Volume 14, Issue 3
A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

Nouh Izem & Mohammed Seaid

Adv. Appl. Math. Mech., 14 (2022), pp. 725-758.

Published online: 2022-02

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  • 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-Lobatto-Legendre 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.

  • Keywords

Discontinuous Galerkin method, well-balanced discretization, Runge-Kutta scheme, multilayer shallow water equations, free-surface flows, mass exchange, wind-driven flows, strait of Gibraltar.

  • AMS Subject Headings

35L65, 35L40, 65M60, 85A30, 86A05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-725, author = {Izem , Nouh and Seaid , Mohammed}, title = {A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {3}, pages = {725--758}, 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-Lobatto-Legendre 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0364}, url = {http://global-sci.org/intro/article_detail/aamm/20282.html} }
TY - JOUR T1 - A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography AU - Izem , Nouh AU - Seaid , Mohammed JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 725 EP - 758 PY - 2022 DA - 2022/02 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0364 UR - https://global-sci.org/intro/article_detail/aamm/20282.html KW - Discontinuous Galerkin method, well-balanced discretization, Runge-Kutta scheme, multilayer shallow water equations, free-surface flows, mass exchange, wind-driven flows, strait of Gibraltar. AB -

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-Lobatto-Legendre 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.

Nouh Izem & Mohammed Seaid. (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. doi:10.4208/aamm.OA-2020-0364
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