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

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. The complexity of the algorithm is also analyzed, showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired accuracy. A WKB analysis is used to prove that the spectral density function is real analytic. Using this new algorithm, we highlight key properties of the partial differential equation and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.