A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems

In this paper, we analyze and design Filon-type asymptotic methods for solving highly oscillatory second-order initial value problems q ″ ( t ) + Mq ( t ) = f ( t , q ( t ) , q ′ ( t ) ) , where M is a non-singular and diagonalizable matrix having large eigenvalues and ‖ M ‖ ≫ 1 . We first derive Filon-type asymptotic methods for highly oscillatory linear systems and then extend the methods to highly oscillatory nonlinear systems. The integrators are a blend of the existing Filon-type methods and asymptotic methods based on the variation-of-constants formula. It is shown that the new integrators are convergent and the accuracy of the new integrators is improved as ‖ M ‖ grows large. This point is significant and powerful for effectively dealing with highly oscillatory systems. Numerical experiments are carried out and the numerical results show the advantage and efficiency of our new integrators as compared with existing numerical methods for highly oscillatory problems from the scientific literature.