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25
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.
A CLASS OF ASYMPTOTIC PRESERVING SCHEMES FOR KINETIC EQUATIONS AND RELATED PROBLEMS WITH STIFF SOURCES
"... Abstract. In this paper, we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by s ..."
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Cited by 12 (3 self)
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Abstract. In this paper, we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGKtype relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGKtype relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptoticpreserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible NavierStokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems,
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
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.
TIME–SPLITTING SCHEMES AND MEASURE SOURCE TERMS For A Quasilinear Relaxing System
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2003
"... Several singular limits are investigated in the context of a 2×2 system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a L² projection operator are concentrated on a discrete lattice by means of Dirac measures. Thi ..."
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Cited by 5 (5 self)
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Several singular limits are investigated in the context of a 2×2 system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a L² projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some timesplitting numerical schemes.
Splitting schemes for the simulation of non equilibrium radiative flows
, 2006
"... Abstract. This paper is devoted to the numerical investigation of radiative hydrodynamics equations. We focus on non–equilibrium regimes and we design asymptotic preserving schemes which can handle the corresponding stiff equations. Our study includes relativistic effects and Doppler corrections. ..."
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Cited by 5 (2 self)
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Abstract. This paper is devoted to the numerical investigation of radiative hydrodynamics equations. We focus on non–equilibrium regimes and we design asymptotic preserving schemes which can handle the corresponding stiff equations. Our study includes relativistic effects and Doppler corrections.
THE RIEMANN PROBLEM FOR THE SHALLOW WATER EQUATIONS WITH DISCONTINUOUS TOPOGRAPHY
, 2007
"... We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves ..."
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Cited by 3 (3 self)
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We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initialvalue problem.
MICROMACRO SCHEMES FOR KINETIC EQUATIONS INCLUDING BOUNDARY LAYERS
, 2012
"... Abstract. We introduce a new micromacro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function f in all its domain as the sum of an equilibrium adapted ..."
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Abstract. We introduce a new micromacro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function f in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with f) and a remaining kinetic part. This equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function. A consequence of this strategy is that no artificial boundary condition is needed in the micromacro models and the exact boundary condition on f is naturally transposed to the macro part of the model. This method provides an ’Asymptotic preserving’ numerical scheme which generates a very good approximation of the space boundary values at the diffusive limit, without any mesh refinement in the boundary layers. Our numerical results are in very good agreement with the exact socalled Chandrasekhar value, which is explicitely known in some simple cases. 1.
Oscillatory behavior of asymptoticpreserving splitting methods for a linear model of diffusive relaxation
 Kinet. Relat. Models
"... (Communicated by Pierre Degond) Abstract. The occurrence of oscillations in a wellknown asymptotic preserving (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the P1 equations. The scheme is derived with operator splitting methods that separat ..."
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(Communicated by Pierre Degond) Abstract. The occurrence of oscillations in a wellknown asymptotic preserving (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the P1 equations. The scheme is derived with operator splitting methods that separate the P1 system into slow and fast dynamics. A careful analysis of the scheme shows that binary oscillations can occur as a result of a blackred diffusion stencil and that dispersivetype oscillations may occur when there is too little numerical dissipation. The latter conclusion is based on comparison with a modified form of the P1 system. Numerical fixes are also introduced to remove the oscillatory behavior.
AN ASYMPTOTIC PRESERVING SCHEME BASED ON A NEW FORMULATION FOR NLS IN THE SEMICLASSICAL LIMIT
, 2012
"... Abstract. We consider the semiclassical limit for the nonlinear Schrödinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists in including some asymptotically vanishing viscosity. We prove that the system is al ..."
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Abstract. We consider the semiclassical limit for the nonlinear Schrödinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists in including some asymptotically vanishing viscosity. We prove that the system is always locally wellposed in a class of Sobolev spaces, and globally wellposed for a fixed positive Planck constant in the onedimensional case. We propose a second order numerical scheme which is asymptotic preserving. Before singularities appear in the limiting Euler equation, we recover the quadratic physical observables as well as the wave function with mesh size and time step independent of the Planck constant. This approach is also well suited to the linear Schrödinger equation.