Uniformly accurate nested Picard integrators for a system of oscillatory ordinary differential equations

Abstract

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter $$0<\varepsilon \le 1$$ . The solution of this system propagates wave with $$O(\varepsilon ^2)$$ wavelength and $$O(\varepsilon ^4)/O(\varepsilon ^2)$$ amplitude for well-prepared/ill-prepared initial data. Based on the NPI and the exponential integrator, a uniformly accurate (w.r.t $$\varepsilon $$ ) second order numerical scheme with $$O(\tau ^2)$$ error has been developed. The method can be generalized to higher order. Numerical tests are presented to confirm our error estimates.