A splitting scheme for the wave maps equation at low regularity

TL;DR

This paper introduces a convergent Lie splitting scheme for the wave maps equation at low regularity, utilizing discrete Bourgain spaces and preserving the null structure to ensure stability.

Contribution

It develops a novel low regularity numerical scheme for wave maps that maintains the null structure at the discrete level, enabling convergence analysis.

Findings

Proves convergence of the scheme for all subcritical initial data in H^s.

Uses discrete Bourgain spaces for low regularity analysis.

Preserves null structure involving time derivatives at the discrete level.

Abstract

We prove convergence of a filtered Lie splitting scheme for the wave maps equation with low regularity initial data in dimension 3. The convergence analysis is performed in discrete Bourgain spaces, as has proved fruitful for the low regularity analysis of the equation in the continuous setting. An important difficulty here is that the analysis of wave maps at low regularity requires the use of the null structure of the system, this structure thus has to be preserved at the discrete level to get an effective stable low regularity scheme. Since the null structure involves time derivatives, the scheme has to be designed carefully. The presence of time derivatives in the nonlinearity then constitutes the most significant source of numerical error. Nonetheless, we are able to prove convergence of the scheme for all subcritical initial data in $H^s$, $s>d/2$.