Dynamical Low-Rank Approximation in Velocity Space for Collisional Kinetic Equations

Solving kinetic equations is computationally demanding due to the high dimensionality of phase space. A direct discretization requires storage and computational complexity on the order of O(n^2d), where n is the number of grid points per dimension and d is the dimension. The Dynamical Low-Rank (DLR) approximation offers a more efficient alternative.

The majority of prior works on the DLR method for kinetic equations apply it to separate the physical and velocity variables. While this approach has achieved success in problems with weak nonlinearities, it does not take advantage of the possible low-rank structure of the solution in velocity space (e.g., the Maxwellian equilibrium is a rank-1 function when viewed in velocity space).

This motivates the development of a velocity-space DLR method, which maintains full resolution in the physical space while exploiting compression in velocity space. We implement this approach for the Vlasov-Fokker-Planck equation, using a Runge-Kutta scheme for time integration and finite-volume methods for spatial discretization. We demonstrate the efficiency and accuracy of the proposed method through numerical experiments on Landau damping, two-stream instability, and the Dory–Guest–Harris (DGH) instability.