(Spectral) Chebyshev collocation methods for solving differential equations

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.

25 pages, 2 figures, 2 tables

References in corpus (2)

• Functionally-fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems
• Arbitrarily high-order energy-conserving methods for Poisson problems

Cited by in corpus (5)

• A spectrally accurate step-by-step method for the numerical solution of fractional differential equations
• Numerical solution of FDE-IVPs by using Fractional HBVMs: the fhbvm code
• Solving FDE-IVPs by using Fractional HBVMs: some experiments with the fhbvm code
• Analysis and implementation of collocation methods for fractional differential equations
• A shooting-Newton procedure for solving fractional terminal value problems