  A general class of KdV-type wave equations regularized with a
convolution-type nonlocality in space is considered. The class differs from the
class of the nonlinear nonlocal unidirectional wave equations previously
studied by the addition of a linear convolution term involving third-order
derivative. To solve the Cauchy problem we propose a semi-discrete numerical
method based on a uniform spatial discretization, that is an extension of a
previously published work of the present authors. We prove uniform convergence
of the numerical method as the mesh size goes to zero. We also prove that the
localization error resulting from localization to a finite domain is
significantly less than a given threshold if the finite domain is large enough.
To illustrate the theoretical results, some numerical experiments are carried
out for the Rosenau-KdV equation, the Rosenau-BBM-KdV equation and a
convolution-type integro-differential equation. The experiments conducted for
three particular choices of the kernel function confirm the error estimates
that we provide.
