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143
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 114 (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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Handbook of Numerical Analysis
"... A consistent intermediate wave speed for a wellbalanced HLLC solver ..."
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Cited by 31 (0 self)
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A consistent intermediate wave speed for a wellbalanced HLLC solver
Numerical Solution of an Inverse Problem in SizeStructured Population Dynamics, in "Inverse Problems
 n o 4, april 2009 BR . Activity Report INRIA 2009
"... We consider a sizestructured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and val ..."
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Cited by 30 (9 self)
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We consider a sizestructured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasireversibility method proposed in [1]. 1
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.
Why many theories of shock waves are necessary. Convergence error in formally pathconsistent schemes
, 2008
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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 21 (4 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.
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 16 (4 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.
2005), On the use of Saint Venant equations to simulate the spreading of a granular mass
 J. Geophys. Res
"... [1] Cliff collapse is an active geomorphological process acting at the surface of the Earth and telluric planets. Recent laboratory studies have investigated the collapse of an initially cylindrical granular mass along a rough horizontal plane for different initial aspect ratios a = Hi/Ri, where Hi ..."
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Cited by 15 (8 self)
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[1] Cliff collapse is an active geomorphological process acting at the surface of the Earth and telluric planets. Recent laboratory studies have investigated the collapse of an initially cylindrical granular mass along a rough horizontal plane for different initial aspect ratios a = Hi/Ri, where Hi and Ri are the initial height and radius, respectively. A numerical simulation of these experiments is performed using a minimal depthintegrated model based on a longwave approximation. A dimensional analysis of the equations shows that such a model exhibits the scaling laws observed experimentally. Generic solutions are independent of gravity and depend only on the initial aspect ratio a and an effective friction angle. In terms of dynamics, the numerical simulations are consistent with the experiments for a 1. The experimentally observed saturation of the final height of the deposit, when normalized with respect to the initial radius of the cylinder, is accurately reproduced numerically. Analysis of the results sheds light on the correlation between the area overrun by the granular mass and its initial potential energy. The extent of the deposit, the final height, and the arrest time of the front can be directly estimated from the ‘‘generic solution’ ’ of the model for terrestrial and extraterrestrial avalanches. The effective friction, a parameter classically used to describe the mobility of gravitational flows, is shown to depend on the initial aspect ratio a. This dependence should be taken into account when interpreting the high mobility of large volume events.
Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation
 J. Comput. Phys
, 2008
"... We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Ty ..."
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Cited by 15 (2 self)
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We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is wellbalanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth nonnegativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque’s wave propagation algorithm [25] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling. Key words: shallow water equations, hyperbolic conservation laws, finite volume methods, Godunov methods, Riemann solvers, wave propagation, shock capturing methods, tsunami modeling