Solving nonlinear ODEs with the ultraspherical spectral method

2023/06/30 by Ouyuan Qin, Kuan Xu, Qin, Ouyuan +1
Engineering · Mathematics · #65F08 #65L10 #65N22 #65N35 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2306.17688

openalex publication_date 2023/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the ultraspherical spectral method to solving nonlinear ODE boundary value problems. We propose to use the inexact Newton-GMRES framework for which an effective preconditioner can be constructed and a fast Jacobian-vector multiplication can be effected, thanks to the structured operators of the ultraspherical spectral method. With a mixed-precision implementation, the inexact Newton-GMRES-ultraspherical framework exhibits extraordinary speed and accuracy, as we show by extensive numerical experiments.

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