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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 127 (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.
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review
, 2010
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From Kinetic Equations to Multidimensional Isentropic Gas Dynamics Before Shocks
, 2003
"... This article is devoted to the proof of the hydrodynamical limit from kinetic equations (including B.G.K. like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method, hence the derivation is valid only before shocks appear on the limit system solution. Howev ..."
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Cited by 32 (4 self)
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This article is devoted to the proof of the hydrodynamical limit from kinetic equations (including B.G.K. like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method, hence the derivation is valid only before shocks appear on the limit system solution. However, no a priori knowledge on high velocities distributions for kinetic functions is needed. The case of the SaintVenant system with topography (where a source term is added) is included. Keywords: Hydrodynamic limit, Entropy method, B.G.K. equation, Isentropic gas dynamics, SaintVenant system.
A SecondOrder WellBalanced Positivity Preserving CentralUpwind Scheme for the SaintVenant System
 Communications in Mathematical Sciences
"... Abstract. A family of Godunovtype centralupwind schemes for the SaintVenant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425]. Depending on the reconstruction step, the secondorder versions of the sche ..."
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Cited by 27 (3 self)
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Abstract. A family of Godunovtype centralupwind schemes for the SaintVenant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425]. Depending on the reconstruction step, the secondorder versions of the schemes there could be made either wellbalanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved secondorder centralupwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one and twodimensional examples. Key words. Hyperbolic systems of conservation and balance laws, semidiscrete centralupwind schemes, SaintVenant system of shallow water equations. AMS subject classifications. 65M99, 35L65 1.
Space Localization And WellBalanced Schemes For Discrete Kinetic Models In Diffusive Regimes
 SIAM J. Numer. Anal
, 2002
"... We derive and study WellBalanced schemes for quasimonotone discrete kinetic models. By means of a rigorous localization procedure, we reformulate the collision terms as nonconservative products and solve the resulting Riemann problem whose solution is selfsimilar. The construction of an Asymptotic ..."
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Cited by 18 (4 self)
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We derive and study WellBalanced schemes for quasimonotone discrete kinetic models. By means of a rigorous localization procedure, we reformulate the collision terms as nonconservative products and solve the resulting Riemann problem whose solution is selfsimilar. The construction of an Asymptotic Preserving (AP) Godunov scheme is straightforward and various compactness properties are established within different scalings. At last, some computational results are supplied to show that this approach is realizable and ecient on concrete 2 &times; 2 models.
A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes
 SIAM J. Sci. Comput
"... Abstract. The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. One of the main interest of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFR ..."
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Cited by 17 (5 self)
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Abstract. The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. One of the main interest of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in a good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by JinXin [Comm. Pure Appl. Math., 45, 235–276 (1995)], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established and thus the nonnegativity of the water height, obtained involving the VFRoe approach, is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order MUSCL extensions are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.
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 14 (6 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.
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.
HighResolution Finite Volume Methods for Dusty Gas, Jets and Plumes”, submitted to
 SIAM J. Sci. Comput
"... Abstract. We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A highreso ..."
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Cited by 7 (1 self)
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Abstract. We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A highresolution wavepropagation algorithm is used to solve the equations numerically. The onedimensional algorithm is shown to give agreement with a shock tube test problem in the literature. The twodimensional algorithm has been applied to model expolsive volcanic eruptions in which an axisymmetric jet of hot dusty gas is injected into the atmosphere and the expected behavior is observed at two different vent velocities. The methodology described here, with extensions to three dimensions and adaptive mesh refinement, is being used for more detailed studies of volcanic jet processes. Key words. Finite volume methods, highresolution methods, volcanic flows, dusty gas, plumes, jets, shocks