A Spectral Transform Method for Singular Sturm-Liouville Problems with Applications to Energy Diffusion in Plasma Physics

Understand

We develop a spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions.

• The solution is represented as a discrete and continuous superposition of normalizable and non-normalizable eigenfunctions via the spectral transform associated with a singular Sturm-Liouville operator.
• We present a new algorithm for computing the spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh-Weyl $m$-function from the complex upper half-plane to the real axis.
• The eigenfunctions and density function are rescaled and a new formula for the limiting value of the $m$-function is derived to avoid amplification of roundoff errors when the solution is reconstructed.