fully discrete scheme is implicit, a nonlinear system of equations has to be solved at each time step. This system is linearized by a simple iterative scheme in which the nonlinear term is lagged backwards in time and the linear part is trivial to invert in Fourier space, as e. g. in [@CK]. The overall method is shown to be of second-order accuracy in $L^{2}$ in space and time.

In the present paper the numerical scheme that we use is a [hybrid]{} finite element-spectral method. We consider the periodic initial-value problem for (\[E11\]) and discretize it in space by the Galerkin method using smooth periodic splines of order $r\geq 3$ on a uniform mesh with meshlength $h$. (Cubic splines, i. e. $r=4$, are mainly used in the computations.) The nonlocal term is computed using a spectral approximation as described in Section \[sec2\]. Then, the system of ode’s representing the semidiscrete problem is discretized in time; we use as a base time-stepping scheme the two-stage, fourth-order accurate, Gauss-Legendre implicit Runge-Kutta method. This scheme has high accuracy and good stability properties and has previously been extensively used for the temporal discretization of stiff partial differential equations with a KdV term, cf. e. 