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48
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 wavepropagation method for conservation laws and balance laws with spatially varying flux functions
 SIAM J. Sci. Comput
, 2002
"... Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a ge ..."
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Cited by 28 (5 self)
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Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A highresolution wavepropagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)− fi−1(Qi−1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be secondorder accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f(q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasisteady problems close to steady state. Key words. finitevolume methods, highresolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
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Cited by 23 (5 self)
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We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
A WellBalanced Scheme Using NonConservative Products Designed for Hyperbolic Systems of Conservation Laws With Source Terms
, 2001
"... The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the ..."
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Cited by 21 (3 self)
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The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the results of DalMaso, LeFloch and Murat [9] about nonconservative products, and the generalized Roe matrixes introduced by Toumi [36] to derive a firstorder linearized wellbalanced scheme in the sense of Greenberg and LeRoux [19]. As a main feature, this approach is able to preserve the right asymptotic behaviour of the original inhomogeneous system [31], which is not a obvious property [6]. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations. Key words: conservation laws, source terms. nonconservative products, balanced scheme. AMS subjects classification: 65M06, 76N15. 1 Current adress: Foundation for Research and Technology Hel...
Wave Propagation Methods for Conservation Laws with Source Terms
 In preparation
, 1998
"... . An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm ..."
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Cited by 12 (3 self)
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. An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm was developed and used to compute nearsteady shallow water flow over variable topography. In this paper we extend this algorithm to nearsteady flow of an ideal gas subject to a static gravitational field. The method is implemented in the software package clawpack. The ability of this method to capture perturbed quasisteady solutions is demonstrated with numerical examples. 1. Introduction We consider the Euler equations in conservation form @ t q +r \Delta f (q) = / (q) (1) where q 2 R m is a vector of conserved quantities, f : R m ! R m is the flux, and / is a source term due to a static gravitational field. It is well known that if f is a nonlinear function of q as for the Eule...
Comparison between threedimensional linear and nonlinear tsunami generation models, Theor. Comput. Fluid Dyn
"... The modeling of tsunami generation is an essential phase in understanding tsunamis. For tsunamis generated by underwater earthquakes, it involves the modeling of the sea bottom motion as well as the resulting motion of the water above it. A comparison between various models for threedimensional wat ..."
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Cited by 12 (6 self)
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The modeling of tsunami generation is an essential phase in understanding tsunamis. For tsunamis generated by underwater earthquakes, it involves the modeling of the sea bottom motion as well as the resulting motion of the water above it. A comparison between various models for threedimensional water motion, ranging from linear theory to fully nonlinear theory, is performed. It is found that for most events the linear theory is sufficient. However, in some cases, more sophisticated theories are needed. Moreover, it is shown that the passive approach in which the seafloor deformation is simply translated to the ocean surface is not always equivalent to the active approach in which the bottom motion is taken into account, even if the deformation is
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on ”Methods and Models of Kinetic Theory
, 2010
"... 2. Hyperbolic systems with stiff relaxations 3 3. Kinetic equations: the Euler regime 8 4. Linear transport equations: the diffusion regime 15 ..."
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Cited by 10 (5 self)
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2. Hyperbolic systems with stiff relaxations 3 3. Kinetic equations: the Euler regime 8 4. Linear transport equations: the diffusion regime 15
Diffusion Limit Of The Lorentz Model: Asymptotic Preserving Schemes
"... This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right ..."
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Cited by 10 (2 self)
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This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right discrete diusion equation with the same value of the diusion coecient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
"... ..."
High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallowwater systems
 Math. Comp
"... Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO r ..."
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Cited by 6 (2 self)
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Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the wellbalanced properties of the resulting schemes. Finally, we will focus on applications to shallowwater systems. 1.