  This paper considers numerical discretization of a nonlocal conservation law
modeling vehicular traffic flows involving nonlocal inter-vehicle interactions.
The nonlocal model involves an integral over the range measured by a horizon
parameter and it recovers the local Lighthill-Richards-Whitham model as the
nonlocal horizon parameter goes to zero. Good numerical schemes for simulating
these parameterized nonlocal traffic flow models should be robust with respect
to the change of the model parameters but this has not been systematically
investigated in the literature. We fill this gap through a careful study of a
class of finite volume numerical schemes with suitable discretizations of the
nonlocal integral, which include several schemes proposed in the literature and
their variants. Our main contributions are to demonstrate the asymptotically
compatibility of the schemes, which includes both the uniform convergence of
the numerical solutions to the unique solution of nonlocal continuum model for
a given positive horizon parameter and the convergence to the unique entropy
solution of the local model as the mesh size and the nonlocal horizon parameter
go to zero simultaneously. It is shown that with the asymptotically
compatibility, the schemes can provide robust numerical computation under the
changes of the nonlocal horizon parameter.
