### 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 language | English (US) |
---|---|

Pages (from-to) | 350-392 |

Number of pages | 43 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 75 |

Issue number | 2 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### 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

**A spectral transform method for singular sturm-liouville problems with applications to energy diffusion in plasma physics.** / Wilkening, Jon; Cerfon, Antoine.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 75, no. 2, pp. 350-392. https://doi.org/10.1137/130941948

}

TY - JOUR

T1 - A spectral transform method for singular sturm-liouville problems with applications to energy diffusion in plasma physics

AU - Wilkening, Jon

AU - Cerfon, Antoine

PY - 2015

Y1 - 2015

N2 - 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.

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.

KW - Continuous spectrum

KW - Fokker-Planck collisions

KW - Spectral density function

KW - Sturm-Liouville theory

KW - Titchmarsh-Weyl m-function

KW - WKB approximation

UR - http://www.scopus.com/inward/record.url?scp=84926456773&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84926456773&partnerID=8YFLogxK

U2 - 10.1137/130941948

DO - 10.1137/130941948

M3 - Article

VL - 75

SP - 350

EP - 392

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -