Efficient Dynamical Low-Rank Approximation for the Vlasov-Amp\`{E}Re-Fokker-Planck System

Abstract

Kinetic equations are difficult to solve numerically due to their high dimensionality. A promising approach for reducing computational cost is the dynamical low-rank algorithm, which decouples the dimensions of the phase space by proposing an ansatz as the sum of separable (rank-1) functions in position and velocity respectively. The fluid asymptotic limit of collisional kinetic equations, obtained in the small-Knudsen number limit, admits a low-rank representation when written as $f = Mg$, where $M$ is the local Maxwellian, and $g$ is low-rank. We apply this decomposition to the Vlasov-Amp\`{e}re-Fokker-Planck equation of plasma dynamics, considering the asymptotic limit of strong collisions and electric field. We implement our proposed algorithm and demonstrate the expected improvement in computation time by comparison to an implementation that evolves the full solution tensor $f$. We also demonstrate that our algorithm can capture dynamics in both the kinetic regime, and in the fluid regime with relatively lower computational effort, thereby efficiently capturing the asymptotic fluid limit.