Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbation

Abstract. A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker-Planck equation is solved numerically using a finite-difference discretisation in physical space, and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker-Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions, hence the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps it is more accurate to solve the random displacement model approximation to the LPDM, rather than use existing schemes designed for long time-steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple "small-noise" scheme of Honeycutt.

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Invariant Manifolds and Long-Time Asymptotics for the Vlasov--Poisson--Fokker--Planck Equation

SIAM Journal on Mathematical Analysis ◽

10.1137/s0036141000371368 ◽

2001 ◽

Vol 33 (2) ◽

pp. 489-507 ◽

Cited By ~ 4

Author(s):

Yoshiyuki Kagei

Keyword(s):

Invariant Manifolds ◽

Planck Equation ◽

Time Asymptotics ◽

Fokker Planck Equation ◽

Long Time ◽

Long Time Asymptotics ◽

Fokker Planck

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A non-local scalar conservation law describing navigation processes

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891620500265 ◽

2020 ◽

Vol 17 (04) ◽

pp. 809-841

Author(s):

Paulo Amorim ◽

Florent Berthelin ◽

Thierry Goudon

Keyword(s):

Conservation Law ◽

Stability Result ◽

Planck Equation ◽

Small Region ◽

Effective Velocity ◽

Fokker Planck Equation ◽

Scalar Conservation Law ◽

Non Local ◽

Scalar Conservation ◽

Fokker Planck

We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.

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