Central-upwind scheme for shallow water equations with discontinuous bottom topography

AbstractWe present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

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Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

Communications in Computational Physics ◽

10.4208/cicp.180513.230514a ◽

2014 ◽

Vol 16 (5) ◽

pp. 1323-1354 ◽

Cited By ~ 5

Author(s):

Manuel Jesús Castro Diaz ◽

Yuanzhen Cheng ◽

Alina Chertock ◽

Alexander Kurganov

Keyword(s):

Shallow Water ◽

Numerical Method ◽

Shallow Water Equations ◽

Numerical Experiments ◽

Operator Splitting ◽

Finite Volume Methods ◽

Splitting Methods ◽

Upwind Scheme ◽

Operator Splitting Methods ◽

Numerical Approaches

AbstractIn this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

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Comparison study of some finite volume and finite element methods for the shallow water equations with bottom topography and friction terms

ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik ◽

10.1002/zamm.200510280 ◽

2006 ◽

Vol 86 (11) ◽

pp. 874-891 ◽

Cited By ~ 10

Author(s):

M. Lukáčová-Medvid'ová ◽

U. Teschke

Keyword(s):

Finite Element ◽

Shallow Water ◽

Finite Volume ◽

Finite Element Methods ◽

Shallow Water Equations ◽

Bottom Topography ◽

Comparison Study

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A stabilized well-balanced RBF-FD meshless method for shallow-water equations

Revista dos Trabalhos de Iniciação Científica da UNICAMP ◽

10.20396/revpibic262018687 ◽

2019 ◽

Author(s):

Joel Antonio Godoy de Moraes ◽

Eduardo Cardoso de Abreu ◽

Luis Guilherme Cunha Santos

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Bottom Topography ◽

Point Clouds ◽

General Criterion ◽

Modified Method ◽

Modified Method Of Characteristics ◽

Polyharmonic Splines ◽

Convection Dominated ◽

Careful Design

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

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