  This paper focuses on the numerical approximation of the solutions of
non-local conservation laws in one space dimension. These equations are
motivated by two distinct applications, namely a traffic flow model in which
the mean velocity depends on a weighted mean of the downstream traffic density,
and a sedimentation model where either the solid phase velocity or the
solid-fluid relative velocity depends on the concentration in a neighborhood.
In both models, the velocity is a function of a convolution product between the
unknown and a kernel function with compact support. It turns out that the
solutions of such equations may exhibit oscillations that are very difficult to
approximate using classical first-order numerical schemes. We propose to design
Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes to
obtain high-order approximations. As we will see, the DG schemes give the best
numerical results but their CFL condition is very restrictive. On the contrary,
FV-WENO schemes can be used with larger time steps. We will see that the
evaluation of the convolution terms necessitates the use of quadratic
polynomials reconstructions in each cell in order to obtain the high-order
accuracy with the FV-WENO approach. Simulations using DG and FV-WENO schemes
are presented for both applications.
