Trigonometrically-fitted method for a periodic initial value problem with two frequencies

Abstract

A second-order differential equation whose solution is periodic with two frequencies has important applications in many scientific fields. Nevertheless, it may exhibit ‘periodic stiffness’ for most of the available linear multi-step methods. The phenomena are similar to the popular Stömer–Cowell class of linear multi-step methods for one-frequency problems. According to the stability theory laid down by Lambert, ‘periodic stiffness’ appears in a two-frequency problem because the production of the step-length and the bigger angular frequency lies outside the interval of periodicity. On the other hand, for a two-frequency problem, even with a small step-length, the error in the numerical solution afforded by a P-stable trigonometrically-fitted method with one frequency would be too large for practical applications. In this paper we demonstrate that the interval of periodicity and the local truncation error of a linear multi-step method for a two-frequency problem can be greatly improved by a new trigonometric-fitting technique. A trigonometrically-fitted Numerov method with two frequencies is proposed and has been verified to be P-stable with vanishing local truncation error for a two-frequency test problem. Numerical results demonstrated that the proposed trigonometrically-fitted Numerov method with two frequencies has significant advantages over other types of Numerov methods for solving the ‘periodic stiffness’ problem.