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We develop a spectrally accurate numerical method to compute solutions of amodel partial differential equation used in plasma physics to describediffusion in velocity space due to Fokker-Planck collisions. The solution isrepresented as a discrete and continuous superposition of normalizable andnon-normalizable eigenfunctions via the spectral transform associated with asingular Sturm-Liouville operator. We present a new algorithm for computing thespectral density function of the operator that uses Chebyshev polynomials toextrapolate the value of the Titchmarsh-Weyl $m$-function from the complexupper half-plane to the real axis. The eigenfunctions and density function arerescaled and a new formula for the limiting value of the $m$-function isderived to avoid amplification of roundoff errors when the solution isreconstructed. The complexity of the algorithm is also analyzed, showing thatthe cost of computing the spectral density function at a point grows lessrapidly than any fractional inverse power of the desired accuracy. A WKBanalysis is used to prove that the spectral density function is real analytic.Using this new algorithm, we highlight key properties of the partialdifferential equation and its solution that have strong implications on theoptimal choice of discretization method in large-scale plasma physicscomputations.

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