A-stable Runge-Kutta methods for semilinear evolution equations - Time-semidiscretizations on scales of Banach spaces

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Banach spaces. We show that a class of implicit, A-stable Runge-Kutta methods which includes Gauss-Legendre collocation methods, when applied to such equations, are smooth as maps from open subsets of the highest scale rung into the lowest scale rung. Moreover, under an additional assumption which is, in particular, satisfied in the Hilbert space case, we prove convergence of the time-semidiscretization Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr'odinger equation on the circle