Error estimates of time-splitting schemes for nonlinear Klein--Gordon equation with rough data

TL;DR

This paper analyzes the convergence and error bounds of time-splitting numerical schemes for the nonlinear Klein-Gordon equation with rough initial data, introducing new techniques for handling low-regularity solutions.

Contribution

It establishes optimal error estimates for Lie and Strang splitting methods under minimal regularity assumptions, utilizing discrete Bourgain spaces for very rough data.

Findings

Optimal error bounds depend sharply on solution regularity.

Discrete Bourgain space technique effectively handles very rough data.

Numerical results confirm theoretical error estimates.

Abstract

In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.