Transport based particle methods for the Fokker-Planck-Landau equation
This work addresses the challenge of simulating the Fokker-Planck-Landau equation for plasma physics or kinetic theory, offering a method that maintains key physical invariants, though it appears incremental as it adapts an existing technique to a specific equation.
The authors tackled the numerical solution of the Landau equation by proposing a particle method inspired by score-based transport modeling, which preserves physical properties like conservation laws and entropy decay. They proved that gradient matching recovers the true solution for Maxwellian molecules and demonstrated numerical experiments in low to moderately high dimensions, comparing it with traditional methods.
We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.