Accurate numerical methods for the collisional motion of (heated) granular flows

Filbet, Francis; Pareschi, Lorenzo; Toscani, Giuseppe

doi:10.1016/j.jcp.2004.06.023

Cited by 19 publications

(33 citation statements)

References 35 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

View full text Add to dashboard Cite

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“…In addition to the unpracticable computational cost of deterministic quadrature rules, the integration has to be handled carefully since it is at the basis of the macroscopic properties of the equation. Additional difficulties are represented by the stiffness induced by the presence of small scales, like the case of small mean free path [25] or the case of large velocities [22].…”

Section: Introductionmentioning

confidence: 99%

Fast algorithms for computing the Boltzmann collision operator

2006

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Abstract. The development of accurate and fast numerical schemes for the five-fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.

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“…It has been recently shown to be convergent by F. Filbet and C. Mouhot in [22] for the elastic case and allows spectral accuracy [12]. It has then been derived for the granular gases operator (3.6) by F. Filbet, L. Pareschi and G. Toscani in [24].…”

Section: Discretization Of the Problemmentioning

confidence: 98%

“…Another approach consists in a direct resolution of the Boltzmann operator on a phase space grid. For instance, deterministic and highly accurate methods based on a spectral discretization of the collisional operator have been proposed by F. Filbet, G. Naldi, L. Pareschi, G. Toscani and G. Russo in [39,37,24] for the space-homogeneous setting. Although being of complexity O(N 2 ), they are spectrally accurate and then need very few points to be precise.…”

Section: Introductionmentioning

confidence: 99%

A rescaling velocity method for dissipative kinetic equations. Applications to granular media

Filbet

Rey

2013

Journal of Computational Physics

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Abstract. We present a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. It is based on the knowledge of the hydrodynamic limit of the model considered, but is able to compute solutions for either dilute or dense regimes. Several applications are presented for Boltzmann-like equations. This method is particularly efficient for numerical simulations of the granular gases equation with dissipative energy: it allows to study accurately the long time behavior of this equation and is very well suited for the study of clustering phenomena.

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Accurate numerical methods for the collisional motion of (heated) granular flows

Cited by 19 publications

References 35 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

Fast algorithms for computing the Boltzmann collision operator

A rescaling velocity method for dissipative kinetic equations. Applications to granular media

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