Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions

Abstract

We study time integration schemes for $\dot H^1$-solutions to the energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show first-order convergence in $L^2$ for the Lie splitting and convergence order $3/2$ for a corrected Lie splitting. To our knowledge this includes the first error analysis performed for scaling-critical dispersive problems. Our approach is based on discrete-time Strichartz estimates, including one (with a logarithmic correction) for the case of the forbidden endpoint. Our schemes and the Strichartz estimates contain frequency cut-offs.