Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time
Provides rigorous numerical analysis for a broad class of semilinear PDEs, ensuring unconditional stability and convergence for practitioners using spectral methods.
The paper proves stability and convergence of A-stable Runge-Kutta time discretizations combined with spectral Galerkin spatial truncation for semilinear evolution equations, with error bounds uniform in spatial resolution and no CFL condition. Results apply to semilinear wave and nonlinear Schrödinger equations.
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.