The KdV equation and the Benjamin-Ono equation are of the form $u_t+uu_x=Au$ where $Au=u_{xxx}$ for the KdV equation and $Au=Hu_{xx}$ ($H$ denotes the Hilbert transform) for the Benjamin-Ono equation. We study operator splitting for these two equations, as well as other related equations. Furthermore, we derive convergent and fully discrete finite difference schemes for the two equations.