Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion
This work provides strong motivation for using Maxwell polynomials in high-dimensional gyrokinetic simulations, but also identifies a critical instability that must be addressed.
The paper studies pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion, finding that Maxwell-based discretizations outperform other schemes by orders of magnitude in most situations, but are subject to a non-modal time stepping instability that requires careful handling.
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.