Filtered Lie-Trotter splitting for the “good” Boussinesq equation: Low regularity error estimates

Abstract:

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 $0<s\leq 2$, overcoming the constraint of $s>1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order $\tau ^{s/2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.