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38
IMEX RungeKutta schemes and hyperbolic systems of conservation laws with stiff diffusive relaxation
 ICNAAM, AIP Conference Proceedings 1168
, 2009
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Asymptoticpreserving schemes for kineticfluid modeling of disperse twophase flows
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An allspeed asymptoticpreserving method for the isentropic Euler and NavierStokes equations
 Commun. Comput. Phys
"... The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach nu ..."
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The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an allspeed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and NavierStokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods, and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system, and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection methodlike incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes. 1
Simulation of fluidparticles flows: heavy particles, flowing regime and asymptoticpreserving schemes
 Commun. Math. Sci
"... Abstract. We are interested in an Eulerian–Lagrangian model describing particulate flows. The model under study consists of the Euler system and a VlasovFokkerPlanck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, ..."
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Abstract. We are interested in an Eulerian–Lagrangian model describing particulate flows. The model under study consists of the Euler system and a VlasovFokkerPlanck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, and lead to a limiting model of purely hydrodynamic type. We design a numerical scheme which is able to capture this asymptotic behavior without requiring prohibitive stability conditions. The construction of this Asymptotic Preserving scheme relies on an implicit discretization of the stiff terms which can be treated by efficient inversion methods. This method is a natural coupling of a kinetic solver for the particles with a kinetic scheme for the hydrodynamic Euler equations. Numerical experiments are conducted to study the performance of this scheme in various asymptotic regimes. Key words. Fluid–particles flows, hydrodynamic regimes, Asymptotic Preserving schemes, Kinetic schemes. AMS subject classifications. 82C80, 82C40, 35L65, 35Q35, 65M06, 76N15, 76M20.
Exponential RungeKutta schemes for inhomogeneous Boltzmann equations with high order of accuracy. arXiv preprint arXiv:1208.2622
, 2012
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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.
A BGKpenalization asymptoticpreserving scheme for the multispecies Boltzmann equation, Numerical Methods for Partial Differential Equations
"... An asymptotic preserving scheme is efficient in solving multiscale problems where both kinetic and hydrodynamic regimes coexist. In this paper we extend the BGKpenalization based asymptotic preserving scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation, to its ..."
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An asymptotic preserving scheme is efficient in solving multiscale problems where both kinetic and hydrodynamic regimes coexist. In this paper we extend the BGKpenalization based asymptotic preserving scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation, to its multispecies counterpart. For the multispecies Boltzmann equation the new difficulties emerge due to: 1) the breaking down of the conservation law for each species; 2) different time scalings of convergence to the equilibriums for disparate masses. We select a suitable local Maxwellian–which is based on the mean velocity and mean temperature–as the penalty function, and justifies various asymptotic properties of this method, for both the multispecies Boltzmann equation and a disparate masses system. This results an asymptoticpreserving scheme for the multispecies Boltzmann equation that can capture the fluid dynamic limit with time step and mesh size much larger than Knudsen number, yet the numerical method does not contain any nonlinear nonlocal implicit solver. Numerical examples demonstrate the correct asymptoticbehavior of the scheme. 1
KINETIC ENTROPY INEQUALITY AND HYDROSTATIC RECONSTRUCTION SCHEME FOR THE SAINTVENANT SYSTEM
"... Abstract. A lot of wellbalanced schemes have been proposed for discretizing the classical SaintVenant system for shallow water flows with nonflat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homog ..."
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Abstract. A lot of wellbalanced schemes have been proposed for discretizing the classical SaintVenant system for shallow water flows with nonflat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example Godunov, Roe, kinetic...). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semidiscrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We prove also that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term. 1.
An asymptoticpreserving Monte Carlo method for the Boltzmann equation,
 J. Comput. Phys.
, 2014
"... Abstract In this work, we propose an asymptoticpreserving Monte Carlo method for the Boltzmann equation that is more efficient than the currently available Monte Carlo methods in the fluid dynamic regime. This method is based on the successive penalty method ..."
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Abstract In this work, we propose an asymptoticpreserving Monte Carlo method for the Boltzmann equation that is more efficient than the currently available Monte Carlo methods in the fluid dynamic regime. This method is based on the successive penalty method