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 ε ∈ (0,1], and their solutions are highly oscillatory in time with wavelength at 𝒪(ε ^2) , which brings severe burdens in numerical computation when ε ≪ 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 ε ∈ (0,1]. Moreover, they are super convergent when the time-step size is smaller than ε2. 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.