2013/10/18 by Jon Wilkening, Antoine Cerfon, Wilkening, Jon +1 · 2 citations
Mathematics · Physics and Astronomy · #34B24 #34L16 #35Q84 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods for differential equations #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1310.5074
openalex publication_date 2013/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a spectrally accurate numerical method to compute solutions of a\nmodel partial differential equation used in plasma physics to describe\ndiffusion in velocity space due to Fokker-Planck collisions. The solution is\nrepresented as a discrete and continuous superposition of normalizable and\nnon-normalizable eigenfunctions via the spectral transform associated with a\nsingular Sturm-Liouville operator. We present a new algorithm for computing the\nspectral density function of the operator that uses Chebyshev polynomials to\nextrapolate the value of the Titchmarsh-Weyl m-function from the complex\nupper half-plane to the real axis. The eigenfunctions and density function are\nrescaled and a new formula for the limiting value of the m-function is\nderived to avoid amplification of roundoff errors when the solution is\nreconstructed. The complexity of the algorithm is also analyzed, showing that\nthe cost of computing the spectral density function at a point grows less\nrapidly than any fractional inverse power of the desired accuracy. A WKB\nanalysis is used to prove that the spectral density function is real analytic.\nUsing this new algorithm, we highlight key properties of the partial\ndifferential equation and its solution that have strong implications on the\noptimal choice of discretization method in large-scale plasma physics\ncomputations.\n