The accurate numerical solution of highly oscillatory ordinary differential equations

Abstract:

An asymptotic theory for weakly nonlinear, highly oscillatory systems of ordinary differential equations leads to methods which are suitable for accurate computation with large time steps. The theory is developed for systems of the form \[ \begin {array}{*{20}{c}} {{\mathbf {Z}}’= (A(t)/\varepsilon ){\mathbf {Z}} + {\mathbf {H}}({\mathbf {Z}},t),} \hfill \\ {{\mathbf {Z}}(0,\varepsilon ) = {{\mathbf {Z}}_0},\quad 0 < t < T,0 < \varepsilon \ll 1,} \hfill \\ \end {array} \] where the diagonal matrix $A(t)$ has smooth, purely imaginary eigenvalues and the components of ${\mathbf {H}}({\mathbf {Z}},t)$ are polynomial in the components of Z with smooth t-dependent coefficients. Computational examples are presented.