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Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation
 J. Comput. Phys
, 2008
"... We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Ty ..."
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We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is wellbalanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth nonnegativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque’s wave propagation algorithm [25] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling. Key words: shallow water equations, hyperbolic conservation laws, finite volume methods, Godunov methods, Riemann solvers, wave propagation, shock capturing methods, tsunami modeling
ON SOME FAST WELLBALANCED FIRST ORDER SOLVERS FOR NONCONSERVATIVE SYSTEMS
"... Abstract. The goal of this article is to design robust and simple first order explicit solvers for onedimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the developm ..."
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Abstract. The goal of this article is to design robust and simple first order explicit solvers for onedimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different oneparameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the wellknown LaxFriedrichs, LaxWendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods. 1.
Noelle, Xing, Shu. Wellbalanced schemes for moving water. 1 High Order Wellbalanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water
"... A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of th ..."
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A characteristic feature of hyperbolic systems of balance laws is the existence of nontrivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of socalled wellbalanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high order accurate, exactly wellbalanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.
RWTH Aachen
"... Multi–layer shallow water equations, Kelvin–Helmholtz instability, adaptive choice of layers. AMS subject classifications: ..."
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Multi–layer shallow water equations, Kelvin–Helmholtz instability, adaptive choice of layers. AMS subject classifications:
arbitrary bed in the presence
"... finitevolume scheme for modeling shallow water flows over an ..."
Contents lists available at ScienceDirect Journal of Computational Physics
"... journal homepage: www.elsevier.com/locate/jcp Wellbalanced and energy stable schemes for the shallow water equations with discontinuous topography ..."
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journal homepage: www.elsevier.com/locate/jcp Wellbalanced and energy stable schemes for the shallow water equations with discontinuous topography