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

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