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)+Mx(t)=f(t,x(t),x?(t)) x ? ( t ) + M x ( t ) = f ( t , x ( t ) , x ? ( t ) ) where high-frequency oscillations are generated by the linear part Mx(t) 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.
filon-type asymptotic methods highly oscillatory differential equations multiple time scales variation-of-constants formula fermipastaulam model