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A new banded Petrov--Galerkin spectral method

Ouyuan Qin, Lu Cheng, Kuan Xu

TL;DR

This work introduces a banded Petrov--Galerkin spectral method for variable-coefficient ODEs by designing recombined trial and test bases from Chebyshev and ultraspherical (and, more generally, Jacobi) polynomials. The key idea is that the resulting matrix $A$ can be written as $A = Q^{T} \Omega L R$, with $L$ formed from a structured combination of differentiation, multiplication, and conversion operators, yielding a strictly banded system that can be assembled and solved in $O(N^2 m n)$ time and $O(n)$ per solve step for fixed $m$ and $N$. The paper shows that Mortensen’s MPG is a special case within a Jacobi-based generalization, unifying existing banded Galerkin approaches and enabling fast construction, including aggressive acceleration of the ultraspherical method. Numerical experiments on a simple variable-coefficient ODE and the Airy equation demonstrate linear-time scaling and substantial speedups over MPG and US, while maintaining spectral accuracy; an overarching Jacobi-based framework also clarifies the relationship among prior methods. The authors provide symbolic tools and code for basis recombination, enabling robust, automated generation of recombined bases under general boundary constraints. The practical impact is a flexible, fast, and unifying approach to sparse Galerkin spectral methods for a broad class of ODEs.

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.

A new banded Petrov--Galerkin spectral method

TL;DR

This work introduces a banded Petrov--Galerkin spectral method for variable-coefficient ODEs by designing recombined trial and test bases from Chebyshev and ultraspherical (and, more generally, Jacobi) polynomials. The key idea is that the resulting matrix can be written as , with formed from a structured combination of differentiation, multiplication, and conversion operators, yielding a strictly banded system that can be assembled and solved in time and per solve step for fixed and . The paper shows that Mortensen’s MPG is a special case within a Jacobi-based generalization, unifying existing banded Galerkin approaches and enabling fast construction, including aggressive acceleration of the ultraspherical method. Numerical experiments on a simple variable-coefficient ODE and the Airy equation demonstrate linear-time scaling and substantial speedups over MPG and US, while maintaining spectral accuracy; an overarching Jacobi-based framework also clarifies the relationship among prior methods. The authors provide symbolic tools and code for basis recombination, enabling robust, automated generation of recombined bases under general boundary constraints. The practical impact is a flexible, fast, and unifying approach to sparse Galerkin spectral methods for a broad class of ODEs.

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.

Paper Structure

This paper contains 14 sections, 3 theorems, 50 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. A banded PG method
  3. Trial and test bases
  4. Banded coefficient matrix
  5. Fast construction of the linear system
  6. Numerical examples
  7. An ODE with simple variable coefficients
  8. Airy Equation
  9. An overarching PG method
  10. Jacobi-based banded PG method
  11. MPG method as a specific instance
  12. Accelerating the US method
  13. Discussion
  14. Mathematica code for basis recombination

Key Result

Theorem 2.1

\newlabellem:banded_cmtx0 For the PG method with trial functions $\phi_{k}(x)$, test functions $\psi_{k}(x)$, and the weight $\omega(x) = (1 - x^2)^{N - 1 / 2}$, the coefficient matrix where and Here, the projection operator $\mathcal{P}_{n+N-1} = \left(I_{n+N-1}, 0_{(n+N-1) \times \infty} \right)$.

Figures (4)

  • Figure 1: Execution times for constructing $M_{k}[a(x)]$ using \ref{['Mlambda']} and the method described in tow1.
  • Figure 1: Solving \ref{['taylor']} with the proposed and MPG methods.
  • Figure 1: Solving the $10$th-order ODE \ref{['tenth']} using the original and the accelerated US methods.
  • Figure 2: Solving \ref{['airy']} for $\epsilon = 10^{-9}$ using the proposed, MPG(R), MPG(NI), and the US methods.

Theorems & Definitions (5)

  • Theorem 2.1
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Lemma 5.1