$O(N^{1/2-r})$ in $H^{1/2}_{p}$. In the same paper the semidiscrete problem was discretized in time in the manner suggested in [@CK], i. e. using as a basis the leap-frog method coupled with implicit Crank-Nicolson differencing of the linear dispersive term. This explicit-implicit time-stepping scheme may be implemented efficiently in Fourier space and does not require solving linear systems of equations; as shown in [@DM1] it has an error bound of $O(N^{1/2-r}+\Delta t^{2})$ in $H^{1/2}_{p}$ under the mild stability condition $N^{1/2}\Delta t\leq C$ for some sufficiently small constant $C$. In addition, in [@DM2] the authors analyze the more efficient spectral collocation method (that was used in actual computations in [@PD1] and elsewhere,) for the BO and ILW equations, and prove that the associated semidiscrete problem converges with an $H^{1/2}_{p}-$error bound of $O(N^{3/2-r})$.

A different type of method for the BO equation was constructed and analyzed in [@TM]. It consists of a Crank-Nicolson time-stepping scheme that is coupled with a spatial discretization in which the nonlinear term is approximated by conservative differencing and the nonlocal term is discretized in physical space by the midpoint quadrature formula, which is then interpreted as a discrete convolution and computed by the discrete Fourier transform. Since the 