Runge-Kutta time semidiscretizations of semilinear PDEs with non-smooth data

TL;DR

This paper analyzes the convergence of Runge-Kutta time discretizations for semilinear PDEs with non-smooth initial data, extending existing results to nonlinear systems and providing sharp error estimates.

Contribution

It extends convergence results for Runge-Kutta methods to semilinear PDEs with non-smooth data, including nonlinear wave and Schrödinger equations, using spectral Galerkin projections.

Findings

Convergence order depends on initial data regularity and method order.

Error estimates are sharp as supported by numerical experiments.

Framework applies to various boundary conditions and PDE types.

Abstract

We study semilinear evolution equations $ \frac {{\rm d} U}{{\rm d} t}=AU+B(U)$ posed on a Hilbert space ${\cal Y}$, where $A$ is normal and generates a strongly continuous semigroup, $B$ is a smooth nonlinearity from ${\cal Y}_\ell = D(A^\ell)$ to itself, and $\ell \in I \subseteq [0,L]$, $L \geq 0$, $0,L \in I$. In particular the one-dimensional semilinear wave equation and nonlinear Schr$\"o$dinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge-Kutta method in time, retaining continuous space, and prove convergence of order $O(h^{p\ell/(p+1)})$ for non-smooth initial data $U^0\in {\cal Y}_\ell$, where $\ell\leq p+1$, for a method of classical order $p$, extending a result by Brenner and Thom$\'e$e for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.