Efficient dynamical low-rank approximation for the Vlasov-Ampère-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-Ampère-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 demonstrate that our algorithm can capture dynamics in the fluid regime with very low rank, thereby efficiently capturing the asymptotic fluid limit. Moreover we find that a modest rank of between 15 and 20 is sufficient to capture kinetic effects on the problems we consider, showing that the approach is applicable and efficient across a range of regimes.