Filtered Lie-Trotter splitting for the "good" Boussinesq equation: low regularity error estimates

We investigate a filtered Lie-Trotter splitting scheme for the ``good" Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in $H^s$ for $0<s\le 2$, overcoming the constraint of $s>1\mathrm{/}2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order ${\tau }^{s\mathrm{/}2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.