A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Nicolas Fournier; Sylvie Méléard

doi:10.1090/S0025-5718-01-01339-4

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Nicolas Fournier
, Sylvie Méléard

IECN/UHP

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

Abstract

Using the main ideas of Tanaka, the measure-solution {P_t}_t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {P_t}_t of solutions {P_t^l}_t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate {P_t^l}_t with the empirical measure {μ_t^l,n}_t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

Original language	English
Pages (from-to)	583-604
Number of pages	22
Journal	Mathematics of Computation
Volume	71
Issue number	238
DOIs	https://doi.org/10.1090/S0025-5718-01-01339-4
Publication status	Published - 1 Jan 2002
Externally published	Yes

Keywords

Boltzmann equations without cutoff
Interacting particle systems
Jump measures
Stochastic differential equations

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10.1090/S0025-5718-01-01339-4

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title = "A stochastic particle numerical method for 3D Boltzmann equations without cutoff",

abstract = "Using the main ideas of Tanaka, the measure-solution \{Pt\}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to \{Pt\}t of solutions \{Ptl\}t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-M{\'e}l{\'e}ard is used and allows us to approximate \{Ptl\}t with the empirical measure \{μtl,n\}t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.",

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T1 - A stochastic particle numerical method for 3D Boltzmann equations without cutoff

AU - Fournier, Nicolas

AU - Méléard, Sylvie

PY - 2002/1/1

Y1 - 2002/1/1

N2 - Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {Ptl}t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate {Ptl}t with the empirical measure {μtl,n}t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

AB - Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {Ptl}t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate {Ptl}t with the empirical measure {μtl,n}t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

KW - Boltzmann equations without cutoff

KW - Interacting particle systems

KW - Jump measures

KW - Stochastic differential equations

U2 - 10.1090/S0025-5718-01-01339-4

DO - 10.1090/S0025-5718-01-01339-4

M3 - Article

AN - SCOPUS:0036013524

SN - 0025-5718

VL - 71

SP - 583

EP - 604

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 238

ER -

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Abstract

Keywords

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