Results 1  10
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19
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.
Localization effects and measure source terms in numerical schemes for balance laws
 Math. Comp
"... Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and ..."
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Cited by 13 (3 self)
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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemannbased numerical schemes. Some illustrative and relevant computational results are provided. 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 12 (3 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 × 2 models.
On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2002
"... This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives. The contents mainly refer to modelling by fluid dynamic and kinetic equations and are arranged in three parts. The first part refers to methodol ..."
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Cited by 12 (0 self)
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This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives. The contents mainly refer to modelling by fluid dynamic and kinetic equations and are arranged in three parts. The first part refers to methodological aspects of mathematical modelling and to the interpretation of experimental results. The second part is devoted to modelling and deals both with methodological aspects and with the description of some specific models. The third part reports about an overview on applications and research perspectives.
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on ā€¯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
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 10 (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.
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 9 (3 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
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 7 (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.
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.