The method of stochastic differential equations for computing the kinetics of a plasma with collisions

Иванов, М. Ф.; Shvets, V. F.

doi:10.1016/0041-5553(80)90313-4

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…(Other technical differences are discussed where appropriate.) Conceptually, our method is more in the spirit of the Particle In Cell (PIC) methods widely used, e.g., in plasma physics [26,14,5,39,1,23,41]. Another interesting numerical method has been recently implemented for the one-dimensional McKean-Vlasov system [6].…”

Section: Introductionmentioning

confidence: 99%

A study of blow-ups in the Keller–Segel model of chemotaxis

Fatkullin¹

2012

Nonlinearity

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We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to resolve and propagate singular solutions. We compare the numerical findings (in two dimensions) with analytical predictions regarding formation and interaction of singularities obtained through analysis of the stochastic differential equations associated with the model.

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Section: Introductionmentioning

confidence: 99%

A study of blow-ups in the Keller–Segel model of chemotaxis

Fatkullin¹

2012

Nonlinearity

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“…In this section, a stochastic differential equation model is formulated for a problem in ion transport. When the system of stochastic differential equations is solved numerically, the resulting numerical method is referred to as the random particle method [2,54,66,114]. A similar procedure is applied in fluid dynamics and is called the random vortex method.…”

Section: Ion Transportmentioning

confidence: 99%

“…, t M . More information on the random particle method and the related random vortex method is available, for example, in references [2,31,54,66,114].…”

Section: Ion Transportmentioning

confidence: 99%

Modeling with Itô Stochastic Differential Equations

Allen¹

2007

Mathematical Modelling: Theory and Applications

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This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed.Manuscripts to be considered for publication lie within the following, nonexhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programming, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural sciences, mathematical biology, mathematical ecology, image processing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractals.Original monographs, comprehensive surveys as well as edited collections will be considered for publication.

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“…From this, the asymptotic cost of the MLMC method is O(ε −2 (ln ε) 2 ) for the Euler-Maruyama method and O(ε −2 ) for the Milstein method. These costs are to be contrasted with the total cost of direct and binary methods, for which K is given by (21). As described in Section 2.3, the computational cost of these methods can be easily calculated by writing the requisite (so that the MSE ≤ ε 2 ) timestep ∆t l and sample size N in terms of ε and substituting directly into (21).…”

Section: Introductionmentioning

confidence: 99%

“…These costs are to be contrasted with the total cost of direct and binary methods, for which K is given by (21). As described in Section 2.3, the computational cost of these methods can be easily calculated by writing the requisite (so that the MSE ≤ ε 2 ) timestep ∆t l and sample size N in terms of ε and substituting directly into (21). For direct methods, the result of doing so is given by (22) so K ∼ O ε −(2+1/α) for a general weak order-α scheme, and K ∼ O(ε −3 ) for the widely used α = 1 direct Euler-Maruyama integration scheme.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Monte Carlo simulation of Coulomb collisions

Rosin

Ricketson

Dimits

et al. 2014

Journal of Computational Physics

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We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau-Fokker-Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε −2 ) or O(ε −2 (ln ε) 2 ), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of O(ε −3 ) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε = 10 −5 . We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

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The method of stochastic differential equations for computing the kinetics of a plasma with collisions

Cited by 8 publications

References 4 publications

A study of blow-ups in the Keller–Segel model of chemotaxis

A study of blow-ups in the Keller–Segel model of chemotaxis

Modeling with Itô Stochastic Differential Equations

Multilevel Monte Carlo simulation of Coulomb collisions

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