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66
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 56 (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.
Hamiltonianpreserving schemes for the Liouville equation with discontinuous potentials
"... When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, We introduce two classes o ..."
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Cited by 30 (25 self)
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When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, We introduce two classes of Hamiltonianpreserving schemes for such problems. By using the constant Hamiltonian across the potential barrier, we introduced a selection criterion for a unique, physically relavant solution to the underlying linear hyperbolic equation with singular coefficients. These scheme have a hyperbolic CFL condition, which is a significant improvement over a conventional discretization. We also establish the positivity, and stability in both l 1 and l ∞ norms, of these discretizations, and conducted numerical experiments to study the numerical accuracy. This work is motivated by the wellbalanced kinetic schemes by Perthame and Simeoni for the shallow water equations with a discontinuous bottom topography, and has applications to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrödinger equation with a discontinuous potential, among other applications.
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 17 (2 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.
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 14 (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.
Godunovtype Approximation for a General Resonant Balance Law With Large Data
, 2003
"... We consider the Cauchy problem for the 2 2 nonstrictly hyperbolic system u t + f(a; u) x g(a; u)a x = 0 (a; u)(0; ) = (a o ; u o ): For possibly large, discontinuous and resonant data, the generalized solution to the Riemann problem is introduced, interaction estimates are carried out using a ne ..."
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Cited by 11 (4 self)
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We consider the Cauchy problem for the 2 2 nonstrictly hyperbolic system u t + f(a; u) x g(a; u)a x = 0 (a; u)(0; ) = (a o ; u o ): For possibly large, discontinuous and resonant data, the generalized solution to the Riemann problem is introduced, interaction estimates are carried out using a new change of variables and the convergence of Godunov approximations is shown. Uniqueness is addressed relying on a suitable extension of Kruzkov's techniques. 1
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 11 (3 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.
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review
, 2012
"... Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schem ..."
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Cited by 10 (5 self)
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Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schemes that are efficient in these asymptotic regimes. The designing principle of AP schemes is to preserve, at the discrete level, the asymptotic limit that drives one (usually the microscopic) equation to its asymptotic (macroscopic) equation. An AP scheme is based on solving the microscopic equation, instead of using a multiphysics approach that couples different physical laws at different scales. When the small scale is not numerically resolved, an AP scheme automatically becomes a macroscopic solver for the limiting equation. The AP methodology offers simple, robust and efficient computational methods for a large class of multiscale kinetic, hyperbolic and other physical problems. This
Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation
 WAVE MOTION
, 2006
"... We develop a numerical scheme for the wave front computation of complete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear w ..."
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Cited by 8 (8 self)
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We develop a numerical scheme for the wave front computation of complete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear waves through inhomogeneous media. The key idea is to incorporate Snell’s Law of Refraction into the numerical flux for the reduced Liouville equation. This scheme allows a hyperbolic CFL condition, under which positivity, and stabilities in both l∞ and l¹ norms, are established. Numerical experiments are carried out to demonstrate the validity and accuracy of this new scheme.
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 8 (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.
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 8 (0 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.