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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 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.
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 12 (7 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.
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 12 (4 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
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
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 9 (2 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.
An asymptotic high order masspreserving scheme for a hyperbolic model of chemotaxis
 SIAM J. Num. Anal
"... Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutio ..."
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Cited by 8 (2 self)
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Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.
A MULTIWAVE APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD BASED ON RELAXATION II NUMERICAL IMPLEMENTATION WITH 3 AND 5 WAVES
"... Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy ..."
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Cited by 8 (4 self)
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Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of [5]. We present a 3wave solver that well resolves fast waves and material contacts, and a 5wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7wave solver, which is highly accurate on all types of waves, will be described in a followup paper. We test the solvers on onedimensional shock tube data and smooth shear waves. (1.1) (1.2)
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
"... ..."
Multiresolution technique and explicitimplicit scheme for multicomponent flows
 J. of Num. Math
, 2005
"... flows ..."
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.