convergence of the error for this experiment computed in the discrete maximum norm and the normalized $L^{2}$ norm at $t=T=1$. Here $N$ is the number of spatial intervals and $M=T/k$. We observe that the spatial rate is practically optimal (four) and that the temporal rate approximates the value $p=3$ as $N, M$ increase. (For this experiment, with the tolerance set at $\epsilon=10^{-10}$, the number of Newton iterations $J_{out}$ came out to be always one and $J_{inn}$ varied in general between one and four provided $k$ and $h$ were sufficiently small.) The theoretical order of accuracy of the two-stage Gauss-Legendre RK method is of course equal to four and this value is observed experimentally for the KdV equation, i. e. when the nonlocal term $\mathcal{G}u_{xx}$ is not present, see e. g. ([@BDKMc], Table 3). In our case, the loss of one order of temporal accuracy is apparently caused by the presence of the nonlocal term: Observe that in the Jacobian $J_{1}(U^{*})$ in the matrix of operators in the left-hand side of (\[E36\]) we did not include the part of the Jacobian $J_{2}=-\Theta_{2}$ corresponding to the nonlocal term but transferred it to the right-hand side, in order to retain sparsity in the 