Exponential Runge Kutta time semidiscetizations with low regularity initial data

Abstract

We apply exponential Runge Kutta time discretizations to semilinear evolution equations $\frac { {\rm d} U}{{\rm d} t}=AU+B(U)$ posed on a Hilbert space ${\mathcal Y}$. Here $A$ is normal and generates a strongly continuous semigroup, and $B$ is assumed to be a smooth nonlinearity from ${\mathcal Y}_\ell = D(A^\ell)$ to itself, and $\ell \in I \subseteq [0,L]$, $L \geq 0$, $0,L \in I$. In particular the semilinear wave equation and nonlinear Schr\"odinger equation with periodic boundary conditions or posed on ${\mathbb R}^d$ fit into this framework. We prove convergence of order $O(h^{\min(\ell,p)})$ for non-smooth initial data $U^0\in{\mathcal Y}_\ell$, where $\ell >0$, for a method of classical order $p$. We show in an example of an exponential Euler discretization of a linear evolution equation that our estimates are sharp, and corroborate this in numerical experiments for a semilinear wave equation. To prove our result we Galerkin truncate the semiflow and numerical method and balance the Galerkin truncation error with the error of the time discretization of the projected system. We also extend these results to exponential Rosenbrock methods.