Implicit Dynamical Tensor Train Approximation for Kinetic Equations with Stiff Fokker--Planck Collisions

TL;DR

This paper introduces an implicit dynamical tensor train method for kinetic equations with stiff Fokker--Planck collisions, improving stability and efficiency over explicit schemes in high-dimensional phase space.

Contribution

It develops an implicit/IMEX low-rank tensor train approach with efficient solvers for Sylvester equations, addressing stiffness in kinetic equations with Fokker--Planck collisions.

Findings

Method scales linearly with velocity grid points.

Achieves high accuracy and efficiency in test problems.

Overcomes stability constraints of explicit schemes.

Abstract

Low-rank methods for kinetic equations have attracted increasing attention due to their effectiveness in reducing the high dimensionality of phase space. In our previous work [G. Wang & J. Hu, J. Comput. Phys. 558 (2026) 114884], we developed a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format, in which explicit time integration is employed in all substeps. As a result, the method is subject to severe stability constraints in the strongly collisional regimes. In this paper, we consider kinetic equations with the (nonlinear) Fokker--Planck collision operator and develop a dynamical low-rank method that employs implicit or implicit-explicit (IMEX) discretizations in appropriate substeps to overcome stiffness. In these implicit substeps, the resulting equations can be formulated as matrix or tensor Sylvester equations, for which we propose efficient direct solvers by exploiting their underlying structure. The overall computational cost of the proposed method scales linearly with respect to the number of grid points in a single velocity dimension, comparable to that of a fully explicit low-rank scheme. We demonstrate the accuracy and efficiency of the proposed method on several representative kinetic test problems.