Asymptotic-numerical solvers for highly oscillatory second-order differential equations

摘要

In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time tau = omega t. The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.