A spectral transform method for singular sturm-liouville problems with applications to energy diffusion in plasma physics

Jon Wilkening, Antoine Cerfon

Research output: Contribution to journalArticle

Abstract

We develop a spectrally accurate numerical method to compute solutions of a model PDE 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 nonnormalizable 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 PDE and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.

Original languageEnglish (US)
Pages (from-to)350-392
Number of pages43
JournalSIAM Journal on Applied Mathematics
Volume75
Issue number2
DOIs
StatePublished - 2015

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Singular Sturm-Liouville Problem
Spectral Density Function
Plasma Physics
Probability density function
Spectral density
M-function
Physics
Transform
Plasmas
Eigenfunctions
Energy
Eigenvalues and eigenfunctions
Mathematical operators
Extrapolate
Sturm-Liouville Operator
Fokker-Planck
Computing
Discretization Method
Rounding error
Chebyshev Polynomials

Keywords

  • Continuous spectrum
  • Fokker-Planck collisions
  • Spectral density function
  • Sturm-Liouville theory
  • Titchmarsh-Weyl m-function
  • WKB approximation

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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title = "A spectral transform method for singular sturm-liouville problems with applications to energy diffusion in plasma physics",
abstract = "We develop a spectrally accurate numerical method to compute solutions of a model PDE 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 nonnormalizable 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 PDE and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.",
keywords = "Continuous spectrum, Fokker-Planck collisions, Spectral density function, Sturm-Liouville theory, Titchmarsh-Weyl m-function, WKB approximation",
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AB - We develop a spectrally accurate numerical method to compute solutions of a model PDE 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 nonnormalizable 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 PDE and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.

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