A multiscale pseudo orbit-averaging algorithm for partial differential equations, with application to gyrokinetic equilibrium in the WHAM mirror.

This work presents an explicit multi-scale algorithm for solving partial differential equations by separating the fast and slow dynamics within a single equation. In cases where the boundaries of dynamical scales are clear, we explore the idea of using alternating phased integrators in time: one phase solves the full system of equations; the other freezes and approximates the fast dynamics to achieve larger time steps. To demonstrate this algorithm, we consider problems related to kinetic plasma simulations of magnetic mirrors, which feature a distinct boundary between a region dominated by the rapid transit losses of particles and a trapped region characterized by slow collisions. Two representative model problems are presented, and the algorithm is implemented in the continuum PDE solver Gkeyll, resulting in a speedup of more than 1,000x compared to a direct integrator for finding steady-state kinetic solutions in the WHAM mirror. Results show strong promise for this algorithm in addressing the issues of simulating other general multiscale phenomena.