Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time

TL;DR

This paper analyzes the stability and convergence of A-stable Runge--Kutta methods combined with spectral Galerkin truncation for semilinear evolution equations, ensuring uniform bounds and error estimates independent of discretization parameters.

Contribution

It establishes regularity, convergence, and error bounds for space-time discretizations of semilinear evolution equations using Galerkin truncation and A-stable Runge--Kutta methods, without coupling time step and spatial resolution.

Findings

Uniform bounds on the Galerkin-truncated semiflow and its discretization.

Convergence of the discretization independent of time step and spatial resolution.

Error estimates depending on initial data smoothness.

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 Schr\"odinger equation.