Stability under Galerkin truncation of Astable RungeKutta discretizations in time
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, Astable RungeKutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkintruncated semiflow and its timediscretization 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 spacetime 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 RungeKutta 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 arXiv:
 arXiv:1007.4712
 Bibcode:
 2010arXiv1007.4712O
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Analysis of PDEs
 EPrint:
 doi:10.1017/S0308210512002028