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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 126 (8 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.
Some approximate Godunov schemes to compute shallowwater equatons with topography
, 2003
"... WestEC here te comput tmp of shallowwath equat ons wi ttN:LE&BN y byFinit Volume metmeN , in a onedimensional framework(tNL&3 allmetB: sint oduced may benatEfiLLN extEfiLL in t o dimensions) . AllmetA:3 are based on a discretcrNB on of tN tBCEAEN(B by a piecewisefunctE n constEC on each ce ..."
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Cited by 81 (5 self)
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WestEC here te comput tmp of shallowwath equat ons wi ttN:LE&BN y byFinit Volume metmeN , in a onedimensional framework(tNL&3 allmetB: sint oduced may benatEfiLLN extEfiLL in t o dimensions) . AllmetA:3 are based on a discretcrNB on of tN tBCEAEN(B by a piecewisefunctE n constEC on each cell of tN mesh, from an original idea of Le Rouxet al. Whereaste WellBalanced scheme of Le Roux is based on tN exact resol utol of each Riemann problem, we consider here approximat Riemann solvers. Several singlestg metleN are derived from tom formalism, and numerical result are comparedt a fract ionalsta metN4 . Some tme cases arepresent& : convergencetn ardsstsN: stsN: in subcritECC and supercriterc configuratfiE s, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefactNL wave over astAA Numerical schemes, combined wi t an appropriat highorder extNfi ion, provideaccurat e and convergent approxim atximN # 2003 Elsevier ScienceLtc Allright s reserved. 1. I5964V139 We stE4 intNL paper some approximat Godunov schemest comput shallowwatN equatlow wit a sourcetur oftBAL4fiN(LA in a onedimensional framework. Allmetfi3L presentL may beextB3L4 natB3L4N t tt tB3L4N(&CL4 al model.
Equilibrium schemes for scalar conservation laws with stiff sources
 Math. Comp
, 2003
"... Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory resu ..."
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Cited by 57 (7 self)
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Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a socalled equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients. 1.
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 19 (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.
APPROXIMATION OF HYPERBOLIC MODELS FOR CHEMOSENSITIVE MOVEMENT
, 2005
"... Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first and secondorder wellbalanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On th ..."
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Cited by 14 (4 self)
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Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first and secondorder wellbalanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a highorder finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the wellbalanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain highorder accuracy. Numerical simulations are performed to verify accuracy and the wellbalanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments.
Relaxation Schemes for the Shallow Water Equations
 INT. J. NUMER. METH. FLUIDS
, 2003
"... We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKut ..."
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Cited by 14 (0 self)
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We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present.
An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, Special issue dedicated to the 70th birthday of Professor ZhongCi Shi
 J. Comput. Math
, 2004
"... Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday We propose a simple numerical method for calculating both unsteady and steady state solution of hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow w ..."
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Cited by 14 (1 self)
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Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday We propose a simple numerical method for calculating both unsteady and steady state solution of hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography, and the quasi onedimensional nozzle flows. We use the interface value, rather than the cellaverages, for the source terms, which results in a wellbalanced scheme that can capture the steady state solution with a remarkable accuracy. This method approximates the source terms via the numerical fluxes produced by an (approximate) Riemann solver for the homogeneous hyperbolic systems with slight additional computation complexity using Newton’s iterations and numerical integrations. This method solves well the subor supercritical flows, and with a transonic fix, also handles well the transonic flows over the concentration. Numerical examples provide strong evidence on the effectiveness of this new method for both unsteady and steady state calculations.
Asymptotic Highorder schemes for integrodifferential problems arising in markets with jumps
, 2006
"... In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore ..."
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Cited by 5 (1 self)
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In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic highorder schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.
Upwinding Sources at Interfaces in Conservation Laws
, 2003
"... Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical ows because of the gravity, and their numerical approximation leads to speci c diculties. In the context of nite volume schemes, many authors have proposed to Upwind Sources at Interfac ..."
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Cited by 3 (0 self)
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Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical ows because of the gravity, and their numerical approximation leads to speci c diculties. In the context of nite volume schemes, many authors have proposed to Upwind Sources at Interfaces, i.e. the \U. S. I." method, while a cellcentered treatment seems more natural. This note gives a general mathematical formalism for such schemes. We de ne consistency and give a stability condition for the \U. S. I." method. We relate the notion of consistency to the \wellbalanced" property, but its stability remains open, and we also study second order approximations as well as error estimates. The general case of a nonuniform spatial mesh is particularly interesting, motivated by two dimensional problems set on unstructured grids.
Numerical FluxSplitting for a Class of Hyperbolic Systems with Unilateral Constraint
, 2003
"... We study in this paper some numerical schemes for hyperbolic systems with unilateral constraint. In particular, we deal with the scalar case, the isentropic gas dynamics system and the fullgas dynamics system. We prove the ..."
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We study in this paper some numerical schemes for hyperbolic systems with unilateral constraint. In particular, we deal with the scalar case, the isentropic gas dynamics system and the fullgas dynamics system. We prove the