Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations

TL;DR

This paper introduces trigonometric collocation methods based on Lagrange basis polynomials for efficiently solving multi-frequency oscillatory second-order differential equations, with proven convergence properties and demonstrated numerical effectiveness.

Contribution

The paper develops a new class of trigonometric collocation methods that are independent of matrix norm and includes a fourth-order scheme for oscillatory systems.

Findings

Methods are convergent regardless of matrix norm.

Numerical experiments confirm high efficiency.

Fourth-order scheme enhances accuracy.

Abstract

In the present work, a kind of trigonometric collocation methods based on Lagrange basis polynomials is developed for effectively solving multi-frequency oscillatory second-order differential equations $q^{\prime\prime}(t)+Mq(t)=f\big(q(t)\big)$. The properties of the obtained methods are investigated. It is shown that the convergent condition of these methods is independent of $\norm{M}$, which is very crucial for solving oscillatory systems. A fourth-order scheme of the methods is presented. Numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper.