Solving nonlinear ODEs with the ultraspherical spectral method

TL;DR

This paper extends the ultraspherical spectral method to nonlinear ODE boundary value problems, leveraging inexact Newton-GMRES with effective preconditioning and mixed-precision computing for high speed and accuracy.

Contribution

It introduces a novel combination of ultraspherical spectral method with inexact Newton-GMRES for nonlinear ODEs, enabling efficient preconditioning and fast Jacobian-vector products.

Findings

Achieves high accuracy with mixed-precision implementation

Demonstrates significant speedup in numerical experiments

Provides effective preconditioning strategies for nonlinear problems

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.