Exponentially weighted Legendre–Gauss Tau methods for linear second-order differential equations

Abstract

We develop a method to solve a class of second-order ordinary differential equations with highly oscillatory solutions. The method consists in combining three different techniques: Legendre–Gauss spectral Tau method, exponential fitting, and coefficient perturbation methods. With our approach, the resulting approximate solutions are expressed in terms of an exponentially weighted Legendre polynomial basis { e ω n x L n ( x ) ; n ≥ 0 } , where ω n are appropriately chosen complex numbers. The accuracy and efficiency of the method are discussed and illustrated numerically.