On a splitting method for the Zakharov system

TL;DR

This paper presents an error analysis of a Lie-Trotter splitting combined with Fourier collocation for the Zakharov system, demonstrating convergence properties and addressing regularity challenges.

Contribution

It provides the first rigorous convergence analysis for this splitting method applied to the Zakharov system, including handling regularity loss.

Findings

First-order convergence in time

High-order convergence in space depending on regularity

CFL-type restriction ensures stability

Abstract

An error analysis of a splitting method applied to the Zakharov system is given. The numerical method is a Lie-Trotter splitting in time that is combined with a Fourier collocation in space to a fully discrete method. First-order convergence in time and high-order convergence in space depending on the regularity of the exact solution are shown for this method. The main challenge in the analysis is to exclude a loss of spatial regularity in the numerical solution. This is done by transforming the numerical method to new variables and by imposing a natural CFL-type restriction on the discretization parameters.