Accurate spectral numerical schemes for kinetic equations with energy diffusion

Jon Wilkening, Antoine Cerfon, Matt Landreman

Research output: Contribution to journalArticle

Abstract

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

Original languageEnglish (US)
Pages (from-to)58-77
Number of pages20
JournalJournal of Computational Physics
Volume294
DOIs
StatePublished - Aug 1 2015

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kinetic equations
polynomials
Polynomials
Kinetics
energy
Eigenvalues and eigenfunctions
eigenvectors
operators
plasma physics
partial differential equations
leaves
Partial differential equations
Physics
grids
continuums
Plasmas
collisions
kinetics

Keywords

  • Continuous spectrum
  • Continuum kinetic calculations
  • Fokker-Plank collisions
  • Orthogonal polynomials
  • Sturm-Liouville theory

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy (miscellaneous)

Cite this

Accurate spectral numerical schemes for kinetic equations with energy diffusion. / Wilkening, Jon; Cerfon, Antoine; Landreman, Matt.

In: Journal of Computational Physics, Vol. 294, 01.08.2015, p. 58-77.

Research output: Contribution to journalArticle

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