RCMS: Right Correction Magnus Series approach for oscillatory ODEs

We consider RCMS, a method for integrating differential equations of the form y^′ = [ λ A + A₁ ( t ) ] y with highly oscillatory solution. It is shown analytically and numerically that RCMS can accurately integrate problems using stepsizes determined only by the characteristic scales of A₁ ( t ), typically much larger than the solution "wavelength". In fact, for a given t grid the error decays with, or is independent of, increasing solution oscillation. RCMS consists of two basic steps, a transformation which we call the right correction and solution of the right correction equation using a Magnus series. With suitable methods of approximating the highly oscillatory integrals appearing therein, RCMS has high order of accuracy with little computational work. Moreover, RCMS respects evolution on a Lie group. We illustrate with application to the 1D Schrödinger equation and to Frenet-Serret equations. The concept of right correction integral series schemes is suggested and right correction Neumann schemes are discussed. Asymptotic analysis for a large class of ODEs is included which gives certain numerical integrators converging to exact asymptotic behaviour.