  We present a second-order accurate numerical method for a class of nonlocal
nonlinear conservation laws called the "nonlocal pair-interaction model" which
was recently introduced by Du, Huang, and LeFloch. Our numerical method uses
second-order accurate reconstruction-based schemes for local conservation laws
in conjunction with appropriate numerical integration. We show that the
resulting method is total variation diminishing (TVD) and converges towards a
weak solution. In fact, in contrast to local conservation laws, our
second-order reconstruction-based method converges towards the unique entropy
solution provided that the nonlocal interaction kernel satisfies a certain
growth condition near zero. Furthermore, as the nonlocal horizon parameter in
our method approaches zero we recover a well-known second-order method for
local conservation laws. In addition, we answer several questions from the
paper from Du, Huang, and LeFloch concerning regularity of solutions. In
particular, we prove that any discontinuity present in a weak solution must be
stationary and that, if the interaction kernel satisfies a certain growth
condition, then weak solutions are unique. We present a series of numerical
experiments in which we investigate the accuracy of our second-order scheme,
demonstrate shock formation in the nonlocal pair-interaction model, and examine
how the regularity of the solution depends on the choice of flux function.
