A method for approximate analysis of a linear evolutionary equation in a Hilbert space

A semidiscrete method for determining an approximate solution of the linear evolutionary equation x'(t) = A(t)x in a Hilbert space with an arbitrary order of convergence to the exact solution is considered. The method is based on time sampling of the original equation with the help of the Magnus expansion of the evolutionary operator and the subsequent approximation of this expansion by rational stability functions associated with the A-stable Runge-Kutta methods.