On the Stability and Accuracy of Clenshaw-Curtis Collocation

TL;DR

This paper analyzes the stability and accuracy of Clenshaw-Curtis collocation methods, showing they are highly A-stable and can achieve accuracy comparable to Gauss-Legendre methods, with practical implications for numerical ODE solutions.

Contribution

It provides closed-form expressions for Runge-Kutta coefficients and demonstrates the high stability and accuracy of Clenshaw-Curtis methods through theoretical analysis and numerical experiments.

Findings

Clenshaw-Curtis methods are A-stable up to many nodes

They can match Gauss-Legendre collocation accuracy

Numerical experiments confirm high stability and accuracy

Abstract

We study the A-stability and accuracy characteristics of Clenshaw-Curtis collocation. We present closed-form expressions to evaluate the Runge-Kutta coefficients of these methods. From the A-stability study, Clenshaw-Curtis methods are A-stable up to a high number of nodes. High accuracy is another benefit of these methods; numerical experiments demonstrate that they can match the accuracy of the Gauss-Legendre collocation, which has the optimal accuracy order of all Runge-Kutta methods.