  In this article, we prove the convergence of a semi-discrete numerical method
applied to a general class of nonlocal nonlinear wave equations where the
nonlocality is introduced through the convolution operator in space. The most
important characteristic of the numerical method is that it is directly applied
to the nonlocal equation by introducing the discrete convolution operator.
Starting from the continuous Cauchy problem defined on the real line, we first
construct the discrete Cauchy problem on a uniform grid of the real line. Thus
the semi-discretization in space of the continuous problem gives rise to an
infinite system of ordinary differential equations in time. We show that the
initial-value problem for this system is well-posed. We prove that solutions of
the discrete problem converge uniformly to those of the continuous one as the
mesh size goes to zero and that they are second-order convergent in space. We
then consider a truncation of the infinite domain to a finite one. We prove
that the solution of the truncated problem approximates the solution of the
continuous problem when the truncated domain is sufficiently large. Finally, we
present some numerical experiments that confirm numerically both the expected
convergence rate of the semi-discrete scheme and the ability of the method to
capture finite-time blow-up of solutions for various convolution kernels.
