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A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows
 SIAM J. Sci. Comput
"... Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when com ..."
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Cited by 42 (4 self)
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Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the socalled wellbalanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a wellbalanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast wellbalanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.
A relaxation scheme for conservation laws with a discontinuous coefficient
 Math. Comp
, 2007
"... Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a wea ..."
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Cited by 33 (6 self)
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Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
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The dynamics of pressureless dust clouds and delta waves
 J. Hyperbolic Differ. Equ
, 1994
"... Abstract. The equations of isothermal gas dynamics are studied in the limit when the sound speed vanishes, giving the socalled pressureless gas equations. The collision of two clouds of dust is modeled with these equations in the case where the clouds have finite extent and are surrounded by vacuum ..."
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Cited by 7 (0 self)
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Abstract. The equations of isothermal gas dynamics are studied in the limit when the sound speed vanishes, giving the socalled pressureless gas equations. The collision of two clouds of dust is modeled with these equations in the case where the clouds have finite extent and are surrounded by vacuum. The delta shock that arises in the initial stage of the collision evolves into a delta rarefactionshock and then into a delta doublerarefaction as first one cloud and then the other is fully accreted into the singularity. A highresolution finite volume method that captures this behavior is also presented and numerical results shown. 1
Solitary Waves in Layered Nonlinear Media
 SIAM J. Appl. Math
, 2003
"... We study longitudinal elastic strain waves in a onedimensional periodicallylayered medium, alternating between two materials with dierent densities and stressstrain relations. If the impedances are dierent, dispersive eects are seen due to reection at the interfaces. When the stressstrain relati ..."
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Cited by 6 (2 self)
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We study longitudinal elastic strain waves in a onedimensional periodicallylayered medium, alternating between two materials with dierent densities and stressstrain relations. If the impedances are dierent, dispersive eects are seen due to reection at the interfaces. When the stressstrain relations are nonlinear, the combination of dispersion and nonlinearity leads to the appearance of solitary waves that interact like solitons. We study the scaling properties of these solitary waves and derive a homogenized system of equations that includes dispersive terms. We show that pseudospectral solutions to these equations agree well with direct solutions of the hyperbolic conservation laws in the layered medium using a highresolution nite volume method. For particular parameters we also show how the layered medium can be related to the Toda lattice, which has discrete soliton solutions.
How to solve systems of conservation laws numerically using the graphics processor as a highperformance computational engine
 Quak (Eds.), Geometric Modelling, Numerical Simulation, and Optimization: Industrial Mathematics at SINTEF
, 2005
"... Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how the ..."
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Cited by 5 (2 self)
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Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how these two schemes fail to give accurate approximations to solutions containing discontinuities. We then introduce a general class of semidiscrete finitevolume schemes that are designed to produce accurate resolution of both smooth and nonsmooth parts of the solution. Using this special class we wish to introduce the reader to the basic principles used to design modern highresolution schemes. As examples of systems of conservation laws, we consider the shallowwater equations for water waves and the Euler equations for the dynamics of an ideal gas. The second theme in the paper is how programmable graphics processor units (GPUs or graphics cards) can be used to efficiently compute numerical solutions of these systems. In contrast to instruction driven microprocessors (CPUs), GPUs subscribe to the datastreambased computing paradigm and have been optimised for high throughput of large data streams. Most modern numerical methods for hyperbolic conservation laws are explicit schemes defined over a grid, in which the unknowns at each grid point or in each grid cell can be updated independently of the others. Therefore such methods are particularly attractive for implementation using datastreambased processing. 1
A SUBSONICWELLBALANCED RECONSTRUCTION SCHEME FOR SHALLOW WATER FLOWS
"... Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reco ..."
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Cited by 5 (3 self)
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Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reconstruction that allows to derive a subsonicwellbalanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semidiscrete entropy inequality. An application to the EulerPoisson system is proposed. 1.
A Wave Propagation Algorithm for Hyperbolic Systems on Curved Manifolds
"... An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, sh ..."
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Cited by 4 (0 self)
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An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and highresolution shockcapturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary onedimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.
High Resolution Methods and Adaptive Refinement for Tsunami Propagation and Inundation.
"... We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents ..."
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Cited by 4 (1 self)
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We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents some unique challenges that must be overcome. We describe some of the specific difficulties associated with tsunami modeling, and summarize some numerical approaches that we have used to overcome these challenges. For instance, we have developed a wellbalanced Riemann solver that is appropriate in the deep ocean regime as well as robust in nearshore and dry regions. Additionally, we have extended adaptive refinement algorithms to this application. We briefly describe some of the modifications necessary for using these adaptive methods for tsunami modeling.