A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

doi:10.1007/s42967-022-00241-x

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (4): 2385-2430.doi: 10.1007/s42967-022-00241-x

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A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

Wasilij Barsukow¹, Jonas P. Berberich²

1 Bordeaux Institute of Mathematics, Bordeaux University and CNRS/UMR5251, 33405 Talence, France;
2 Wurzburg University, Emil-Fischer-Strasse 40, 97074 Würzburg, Germany

收稿日期:2022-01-23 修回日期:2022-08-11 接受日期:2022-12-01 发布日期:2024-12-20
通讯作者: Wasilij Barsukow,E-mail:wasilij.barsukow@math.u-bordeaux.fr E-mail:wasilij.barsukow@math.u-bordeaux.fr
基金资助:
Wasilij Barsukow was supported by the Deutsche Forschungsgemeinschaft (DFG) through project 429491391(BA 6878/1-1).Jonas P.Berberich was supported by the Klaus Tschira Foundation.

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

Wasilij Barsukow¹, Jonas P. Berberich²

1 Bordeaux Institute of Mathematics, Bordeaux University and CNRS/UMR5251, 33405 Talence, France;
2 Wurzburg University, Emil-Fischer-Strasse 40, 97074 Würzburg, Germany

Received:2022-01-23 Revised:2022-08-11 Accepted:2022-12-01 Published:2024-12-20
Contact: Wasilij Barsukow,E-mail:wasilij.barsukow@math.u-bordeaux.fr E-mail:wasilij.barsukow@math.u-bordeaux.fr

摘要/Abstract

摘要： Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

关键词: Finite volume methods, Active Flux, Shallow water equations, Dry states, Well-balanced methods

Abstract: Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

Key words: Finite volume methods, Active Flux, Shallow water equations, Dry states, Well-balanced methods

Wasilij Barsukow, Jonas P. Berberich. A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying[J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2385-2430.

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[1]	Wasilij Barsukow. Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids[J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 638-652.
[2]	R. Abgrall. A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 370-402.

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

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