A new banded Petrov--Galerkin spectral method

TL;DR

This paper introduces a novel banded Petrov-Galerkin spectral method for solving ODEs with variable coefficients, offering computational efficiency, a unifying framework for existing methods, and improved performance through acceleration techniques.

Contribution

The paper presents a new banded Petrov-Galerkin spectral method that efficiently handles variable coefficients and unifies existing banded spectral methods, addressing key challenges in basis design and sparsity.

Findings

Achieves linear complexity in constructing and solving linear systems.

Demonstrates superior performance compared to Mortensen's and ultraspherical spectral methods.

Provides a systematic approach for designing recombined bases.

Abstract

We propose a Petrov--Galerkin spectral method for ODEs with variable coefficients. When the variable coefficients are smooth, the new method yields a strictly banded linear system, which can be efficiently constructed and solved in linear complexity. The performance advantage of our method is demonstrated through benchmarking against Mortensen's Galerkin method and the ultraspherical spectral method. Furthermore, we introduce a systematic approach for designing the recombined basis and establish that our new method serves as a unifying framework that encompasses all existing banded Galerkin spectral methods. This significantly addresses the ongoing challenge of developing recombined bases and sparse Galerkin spectral method. Additionally, the accelerating techniques presented in this paper can also enhance the performance of the ultraspherical spectral method.