Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations
This work provides a new numerical method for solving oscillatory systems in computational science, where convergence independent of the matrix norm is crucial for stability.
The authors developed trigonometric collocation methods using Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations, achieving convergence independent of the matrix norm and demonstrating remarkable efficiency through numerical experiments.
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.