(Spectral) Chebyshev collocation methods for solving differential equations

TL;DR

This paper explores the use of Chebyshev polynomial-based collocation methods for solving differential equations, extending previous Legendre-based approaches to improve spectral accuracy and analyze new method generalizations.

Contribution

It introduces a novel framework for Chebyshev collocation methods, generalizing existing Hamiltonian Boundary Value Methods and providing new analysis and extensions.

Findings

Enhanced spectral accuracy in solving differential equations.

Generalized Chebyshev collocation methods with improved properties.

Theoretical analysis of new Chebyshev-based numerical schemes.

Abstract

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.