Numerical solution of the two-layer shallow water equations with bottom topography

2002 ◽  
Vol 60 (4) ◽  
pp. 605-638 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Numerical Solution ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Bottom Topography

scholarly journals Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

2011 ◽  
Vol 10 (2) ◽  
pp. 371-404 ◽  
Author(s):  
Andreas Bollermann ◽  
Sebastian Noelle ◽  
Maria Lukáčová-Medvid’ová
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Main Idea ◽  
Rarefaction Waves ◽  
Finite Volume Schemes ◽  
A Cell ◽  
Gravity Driven ◽  
Water Height

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.

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

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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