Wulff's paper accepted in Proc Royal Society of Edinburgh

The paper “Stability under Galerkin truncation of A-stable Runge-Kutta semidiscretizations in time“, by Marcel Oliver and Claudia Wulff has been accepted for publication in the Proceedings of the Royal Society of Edinburgh, Section A: Mathematics. The paper considers semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. Their semiflow is approximated by an implicit, A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space.  Regularity is shown of the Galerkin-truncated semiflow and its time-discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value.  Also proved is convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. Then the Galerkin truncation error is estimated for the semiflow of the evolution equation, its Runge–Kutta discretization, and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. The results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation.  A copy of the final form preprint can be downloaded here.