Spectral Analysis of Continuum Kinetic Velocity Diffusion
Jon Wilkening, UC Berkeley
A promising idea for reducing the cost of continuum kinetic
calculations modeling Fokker-Planck collisions in plasma physics is
represent the speed coordinate using non-standard orthogonal
polynomials. However, traditional pseudo-spectral
this basis have been found to be unstable. We propose three
alternatives in the context of a model PDE that describes diffusion
First, to understand the mathematical structure of the PDE, we have
developed a new algorithm for computing the spectral density
of singular Sturm-Liouville operators. This leads to a
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
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)
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
problem and agrees to roundoff accuracy with the Galerkin approach.
This is joint work with Antoine Cerfon.