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14
Asymptoticpreserving & wellbalanced schemes for radiative transfer and the Rosseland approximation
, 2003
"... We are concerned with efficient numerical simulation of the radiative transfer equations... ..."
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Cited by 21 (3 self)
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We are concerned with efficient numerical simulation of the radiative transfer equations...
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.
A WELLBALANCED NUMERICAL SCHEME FOR A ONE DIMENSIONAL QUASILINEAR HYPERBOLIC MODEL OF
, 2012
"... Abstract. We introduce a numerical scheme to approximate a quasilinear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum a ..."
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Cited by 2 (0 self)
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Abstract. We introduce a numerical scheme to approximate a quasilinear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum and, besides, which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system. AMS Primary: 65M08; Secondary: 35L60, 92B05, 92C17. 1.
MODEL ADAPTATION FOR HYPERBOLIC SYSTEMS WITH RELAXATION
, 2013
"... Abstract. In numerous applications, a hierarchy of models is available to describe the phenomenon under consideration. We focus in this work on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. The goal is to determine the r ..."
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Abstract. In numerous applications, a hierarchy of models is available to describe the phenomenon under consideration. We focus in this work on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. The goal is to determine the regions of the computational domain where the relaxation model (socalled fine model) can be replaced by the equilibrium model (socalled coarse model), in order to simplify the computation while keeping the global numerical accuracy. With this goal in mind, a numerical indicator which measures the difference between the solutions of both models is developed, using a numerical ChapmanEnskog expansion. The reliability of the adaptation procedure is assessed on various test cases coming from twophase flow modeling. Keywords. Hyperbolic system, finite volume methods, relaxation, model adaptation, ChapmanEnskog expansion, twophase flows. Mathematics Subject Classification. 35L45, 65M08, 65M55, 35C20, 76T10 1.
On the Asymptotic Preserving property of the Unified Gas Kinetic Scheme for the diffusion limit of linear kinetic models
, 2013
"... Abstract. The unified gas kinetic scheme (UGKS) of K. Xu et al. [37], originally developed for multiscale gas dynamics problems, is applied in this paper to a linear kinetic model of radiative transfer theory. While such problems exhibit purely diffusive behavior in the optically thick (or small Knu ..."
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Abstract. The unified gas kinetic scheme (UGKS) of K. Xu et al. [37], originally developed for multiscale gas dynamics problems, is applied in this paper to a linear kinetic model of radiative transfer theory. While such problems exhibit purely diffusive behavior in the optically thick (or small Knudsen) regime, we prove that UGKS is still asymptotic preserving (AP) in this regime, but for the free transport regime as well. Moreover, this scheme is modified to include a time implicit discretization of the limit diffusion equation, and to correctly capture the solution in case of boundary layers. Contrary to many AP schemes, this method is based on a standard finite volume approach, it does neither use any decomposition of the solution, nor staggered grids. Several numerical tests demonstrate the properties of the scheme. Key words. Transport equations, diffusion limit, asymptotic preserving schemes, stiff terms 1
HYDRODYNAMIC LIMITS FOR KINETIC EQUATIONS AND THE DIFFUSIVE APPROXIMATION OF RADIATIVE TRANSPORT FOR ACOUSTIC WAVES
"... Abstract. We consider a class of kinetic equations equipped with a single conservation law which generate L 1contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the “dissipativ ..."
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Abstract. We consider a class of kinetic equations equipped with a single conservation law which generate L 1contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the “dissipative ” sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in L 1. The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation. 1.
x Radiation Hydrodynamics, Modelization and Asymptotic Preserving Schemes
"... In view of radiation hydrodynamics computations, we propose an implicit numerical scheme that captures the diffusion limit of the two moment approximate model for the radiative transfer. We prove by construction the limited flux property. Various test cases show the accuracy and robustness of the sc ..."
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In view of radiation hydrodynamics computations, we propose an implicit numerical scheme that captures the diffusion limit of the two moment approximate model for the radiative transfer. We prove by construction the limited flux property. Various test cases show the accuracy and robustness of the scheme. Key words: radiation hydrodynamics, diffusion limit, flux limited and monotone scheme. 1
A SEMICLASSICAL COUPLED MODEL FOR THE TRANSIENT SIMULATION OF SEMICONDUCTOR DEVICES
"... Abstract. We consider the approximation of a microelectronic device corresponding to a n + − n − n + diode consisting in a channel flanked on both sides by two highly doped regions. This is modelled through a system of equations: ballistic for the channel and driftdiffusion elsewhere. The overall c ..."
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Abstract. We consider the approximation of a microelectronic device corresponding to a n + − n − n + diode consisting in a channel flanked on both sides by two highly doped regions. This is modelled through a system of equations: ballistic for the channel and driftdiffusion elsewhere. The overall coupling stems from the Poisson equation for the selfconsistent potential. We propose an original numerical method for its processing, being realizable, explicit in time and nonnegativity preserving on the density. In particular, the boundary conditions at the junctions express the continuity of the current and don’t destabilize the general scheme. At last, efficiency is shown by presenting results on testcases of some practical interest. Key words. Driftdiffusion equation, SchrödingerPoisson equation, WKB ansatz, open quantum system, Robin boundary condition. AMS subject classifications. 76W05, 65J10. hal00426854, version 1 28 Oct 2009
Part 1: The case of the time dependent P1 equations.
, 2008
"... of the time dependent Pn equations by a Godunov type scheme having the diffusion limit. ..."
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of the time dependent Pn equations by a Godunov type scheme having the diffusion limit.