Integral propagator solvers for Vlasov–Fokker–Planck equations

Abstract

We briefly discuss the use of short-time integral propagators on solving the so-called Vlasov–Fokker–Planck equation for the dynamics of a distribution function. For this equation, the diffusion tensor is singular and the usual Gaussian representation of the short-time propagator is no longer valid. However, we prove that the path-integral approach on solving the equation is, in fact, reliable by means of our generalized propagator, which is obtained through the construction of an auxiliary solvable Fokker–Planck equation. The new representation of the grid-free advancing scheme describes the inherent cross- and self-diffusion processes, in both velocity and configuration spaces, in a natural manner, although these processes are not explicitly depicted in the differential equation. We also show that some splitting methods, as well as some finite-difference schemes, could fail in describing the aforementioned diffusion processes, governed in the whole phase space only by the velocity diffusion tensor. The short-time transition probability offers a stable and robust numerical algorithm that preserves the distribution positiveness and its norm, ensuring the smoothness of the evolving solution at any time step.

Self-supervised Learning Method to Solve the Inverse Problem of Fokker-Planck Equation

“…In all these applications on real-world pdfs, the FPE terms are proposed first and then validated by comparing the calculated pdf with the experimental data, which can be seen as a forward problem. To date, many numerical approaches have been proposed to solve the forward problem of FPE which is to calculate the pdf with given drift and diffusion terms of FPE, including the finite element method [8][9][10] , the finite difference method 11,12 , and the path integral method 13 .…”

Propagator integral method for plasma physics kinetic equations with practical applications

“…As it is shown in chapter 3, these equations are, but not restricted to drift-diffusion, purely convective and the Poisson's equation. The first type of equations has been widely dealt in the past by our research group [21][22][23][24][25][26][41][42][43][44] and by other authors [29, 31-33, 39, 40]. Here, a general procedure to obtain short-time solutions as propagators, including also the possible boundary effects, is revisited and expanded.…”

“…For this equation, also known as the Klein-Kramers equation [62][63][64], many works have applied integral methods in an open space [35,41,44,65] providing good result and capturing dynamics that classical schemes would miss, e.g., the parasite effective diffusion in the q 1 q 1 and q 1 q 2 transversal directions. In here, the PIM is used to solve this type of equations in an open domain.…”

“…is the short-time propagator by a double Fourier transformation as 34) reduces to ∂P ∂τ 35) that can be solved by the Method of Characteristics dealing withP = 1 for τ = 0 [41,44], givenP 36) which after Fourier inversion provides 37) where…”

“…An interesting first feature is that even when the diffusion process only appears in the direction v, the resolution of the auxiliary problem shows that D vv produces diffusion in the directions x and in x-v . Each diffusive effect evolves at different time scales (τ −1 , τ −3 and τ −2 respectively), a fact that is missed in a classical difference discretisations [44], which can give rise to non-physical evolutions of f . Thanks to the ability of the PIM to advance f as a semi-analytical approximated solution, these scales, which really appear in an analytical Brownian motion solution, are properly represented.…”