Improved Filon-type asymptotic methods for highly oscillatory differential equations with multiple time scales

In this paper we consider multi-frequency highly oscillatory second-order differential equations ^{x ″}(t) + M x(t) = f(t, x(t), ^{x '}(t)) where high-frequency oscillations are generated by the linear part M x(t), and M is positive semi-definite (not necessarily nonsingular). It is known that Filon-type methods are effective approach to numerically solving highly oscillatory problems. Unfortunately, however, existing Filon-type asymptotic methods fail to apply to the highly oscillatory second-order differential equations when M is singular. We study and propose an efficient improvement on the existing Filon-type asymptotic methods, so that the improved Filon-type asymptotic methods can be able to numerically solving this class of multi-frequency highly oscillatory systems with a singular matrix M. The improved Filon-type asymptotic methods are designed by combining Filon-type methods with the asymptotic methods based on the variation-of-constants formula. We also present one efficient and practical improved Filon-type asymptotic method which can be performed at lower cost. Accompanying numerical results show the remarkable efficiency.