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M. Oliver and C. Wulff,
Stability under Galerkin truncation of A-stable Runge-Kutta
discretizations in time,
P. Roy. Soc. Edinb. A 144 (2014), 603-636.
Abstract:
We consider 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.
We approximate their semiflow by an implicit, A-stable Runge-Kutta
discretization in time and a spectral Galerkin truncation in space.
We show regularity 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. We also
prove convergence of the space-time discretization without any
condition that couples the time step to the spatial resolution. Then
we estimate the Galerkin truncation error 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. Our
results apply, in particular, to the semilinear wave equation and to
the nonlinear Schrödinger equation.
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