A semi-discrete first-order low regularity exponential integrator for the "good" Boussinesq equation without loss of regularity

TL;DR

This paper introduces a novel low regularity exponential integrator for the 'good' Boussinesq equation, achieving first-order accuracy without additional regularity assumptions, confirmed by numerical experiments.

Contribution

It proposes the first low regularity integrator that attains optimal first-order convergence for the GB equation without regularity loss.

Findings

Converges linearly in $H^r$ for solutions in $H^{r+p(r)}$

Achieves first-order accuracy in $H^r$ with no extra derivatives for $r>5/2$

Numerical experiments confirm theoretical convergence

Abstract

In this paper, we propose a semi-discrete first-order low regularity exponential-type integrator (LREI) for the ``good" Boussinesq equation. It is shown that the method is convergent linearly in the space $H^r$ for solutions belonging to $H^{r+p(r)}$ where $0\le p(r)\le 1$ is non-increasing with respect to $r$, which means less additional derivatives might be needed when the numerical solution is measured in a more regular space. Particularly, the LREI presents the first-order accuracy in $H^{r}$ with no assumptions of additional derivatives when $r>5/2$. This is the first time to propose a low regularity method which achieves the optimal first-order accuracy without loss of regularity for the GB equation. The convergence is confirmed by extensive numerical experiments.