An explicit multiscale pseudo orbit-averaging time integration algorithm

TL;DR

This paper introduces an explicit multiscale time integration algorithm that efficiently solves differential equations with high-frequency modes by separating and averaging fast and slow dynamics, achieving significant computational speedups.

Contribution

The paper presents a phased time integrator for multiscale problems, enabling explicit solutions with large speedups in plasma kinetic models by separating fast and slow dynamics.

Findings

Achieved a speedup of order ω/ν_c in model problems.

Demonstrated a 30,000× speedup in a practical plasma simulation.

Validated the algorithm on reduced kinetic models of plasmas.

Abstract

We present an explicit multiscale algorithm for solving differential equations for problems with high-frequency modes that can be averaged over by separating and scaling the fast and slow dynamics within a single equation. We introduce a phased time integrator for cases where the boundaries of dynamical scales are known: one phase solves the unmodified equation, while the other freezes part of phase-space and slows down the evolution of the fast dynamics. This algorithm is applied to reduced kinetic models of plasmas in magnetic mirrors, which feature a distinct boundary between a region dominated by rapid particle transit and a region characterized by slow collisions. Two representative model problems are presented that decompose the dynamics of the magnetic mirror into a simpler, computationally inexpensive form. The model problems demonstrate a speedup by a factor of order $\omega / \nu_c$, where $\omega$ is the fast oscillation frequency and $\nu_c$ is the slow damping rate. This is a 30,000$\times$ speedup for a case of practical interest.