Implicit Dynamical Tensor Train Approximation for Kinetic Equations with Stiff Fokker--Planck Collisions
This work addresses the stability bottleneck in low-rank simulations of kinetic equations with stiff collisions, enabling efficient computation for plasma physics and rarefied gas dynamics.
The authors developed an implicit dynamical low-rank method for kinetic equations with stiff Fokker-Planck collisions, overcoming stability constraints of explicit methods. The method achieves linear computational cost in velocity grid points and demonstrates accuracy on test problems.
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.