Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion

TL;DR

This paper investigates the use of Maxwell polynomial-based pseudo spectral collocation for discretizing kinetic equations with energy diffusion, demonstrating superior accuracy but highlighting stability challenges that require careful operator discretization.

Contribution

It introduces a Maxwell polynomial-based spectral discretization method for kinetic equations and analyzes its stability and performance, showing significant improvements over existing schemes.

Findings

Maxwell polynomial discretizations outperform traditional schemes in accuracy.

The method exhibits non-modal time stepping instability without proper operator discretization.

Careful linear operator representation is essential to leverage the method's advantages.

Abstract

We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional equation describing energy diffusion due to Fokker-Planck collisions with a Maxwell-Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.