# RCMS: right correction Magnus series approach for oscillator ODEs

# Ilan Degani and Jeremy Schiff

Journal of Computational and Applied Mathematics **193** 413-436 (2006).

We consider RCMS, a method for integrating differential equations of the
form *y'=[λA+A*_{1}(t)]yA_{1}(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 Schrodinger 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.