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M. Oliver and C. Wulff,
A-stable Runge-Kutta methods for semilinear
evolution equations
J. Functional Anal. 263 (2012), 1981-2023.
Abstract:
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 Hilbert spaces. In this setting, we
prove the existence of solutions which are temporally smooth in the
norm of the lowest rung of the scale for an open set of initial data
on the highest rung of the scale. Under the same assumptions, we prove
that a class of implicit, A-stable Runge-Kutta semidiscretizations
in time of such equations are smooth as maps from open subsets of the
highest rung into the lowest rung of the scale. Under the additional
assumption that the linear part of the evolution equation is normal or
sectorial, we prove full order convergence of the semidiscretization
in time for initial data on open sets. Our results apply, in
particular, to the semilinear wave equation and to the nonlinear
Schrödinger equation.
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