High order numerical methods for the space non-homogeneous Boltzmann equation

Filbet, Francis; Russo, Giovanni

doi:10.1016/s0021-9991(03)00065-2

Cited by 102 publications

(95 citation statements)

References 22 publications

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“…We have implemented the first order (2.6) and second order (2.8) scheme for the approximation of the Boltzmann equation. Here, the Boltzmann collision operator is discretized by a deterministic method [22][23][24][25]27,52], which gives a spectrally accurate approximation. A classical second order finite volume scheme with slope limiters is applied for the transport operator as sdescribed in Section 4.4.…”

Section: Numerical Testsmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

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a b s t r a c tIn 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 BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving 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 Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

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Section: Numerical Testsmentioning

confidence: 99%

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

Filbet

Jin

2010

Journal of Computational Physics

Self Cite

320

432

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“…It is not as usual for Boltzmann solvers, which usually are based on particle methods which require a grid only in the space variables, but grid based deterministic or semi-deterministic schemes are also available, see [20] or [28] and references therein. It is also possible to use other deterministic methods as Boltzmann solvers, as [28] or [16].…”

Section: Discussionmentioning

confidence: 99%

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

Alaia

Puppo

2012

Journal of Computational Physics

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In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.

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“…Let us first mention that in [18,19] a similar problem, the trend to equilibrium of the solution to the nonhomogeneous Boltzmann equation, is investigated numerically. It is shown that the relative entropy with respect to the local equilibrium oscillates with time when the solution becomes close to the equilibrium.…”

Section: Numerical Simulationsmentioning

confidence: 99%

An asymptotically stable scheme for diffusive coagulation-fragmentation models

Filbet¹

2008

Communications in Mathematical Sciences

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Abstract. This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation with diffusion in space. A finite volume scheme is developed, based on a conservative formulation of the spatially nonhomogeneous coagulation-fragmentation model. It is shown that the scheme preserves positivity, total volume and global steady states. Finally, several numerical simulations are performed to investigate the long time behavior of the solution.

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High order numerical methods for the space non-homogeneous Boltzmann equation

Cited by 102 publications

References 22 publications

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

An asymptotically stable scheme for diffusive coagulation-fragmentation models

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