Spectral Analysis of Continuum Kinetic Velocity Diffusion Equations
Jon Wilkening, UC Berkeley

A promising idea for reducing the cost of continuum kinetic calculations modeling Fokker-Planck collisions in plasma physics is to
represent the speed coordinate using non-standard orthogonal polynomials.  However, traditional pseudo-spectral discretizations in
this basis have been found to be unstable.  We propose three viable alternatives in the context of a model PDE that describes diffusion in
velocity space.

First, to understand the mathematical structure of the PDE, we have developed a new algorithm for computing the spectral density function
of singular Sturm-Liouville operators.  This leads to a generalized Fourier transform in which the solution of the PDE is represented at
each time as a continuous superposition of (non-normalizable) eigenfunctions.  Second, we show how to solve the projected dynamics
in spaces of orthogonal polynomials using a pure Galerkin approach. We compare the spectral density solution to the projected dynamics
solution and find that for a large class of (mildly singular) initial conditions, the new orthogonal polynomials can be 10 orders of
magnitude more accurate than classical Hermite polynomials for the same computational work. Finally, we present a new pseudo-spectral
collocation method that respects the Sturm-Liouville structure of the problem and agrees to roundoff accuracy with the Galerkin approach.

This is joint work with Antoine Cerfon.