On nested Picard iterative integrators for highly oscillatory second-order differential equations

This paper is devoted to the construction and analysis of uniformly accurate (UA) nested Picard iterative integrators (NPI) for highly oscillatory second-order differential equations. The equations involve a dimensionless parameter epsilon is an element of (0, 1], and their solutions are highly oscillatory in time with wavelength at O(epsilon(2)), which brings severe burdens in numerical computation when epsilon << 1. In this work, we first propose two NPI schemes for solving a differential equation. The schemes are uniformly first- and second-order accurate for all epsilon is an element of (0, 1]. Moreover, they are super convergent when the time-step size is smaller than epsilon(2). Then, the schemes are generalized to a system of differential equations with the same uniform accuracies. Error bounds are rigorously established and numerical results are reported to confirm the error estimates.