  We propose an upwind finite volume method for a system of two kinetic
equations in one dimension that are coupled through nonlocal interaction terms.
These cross-interaction systems were recently obtained as the mean-field limit
of a second-order system of ordinary differential equations for two interacting
species. Models of this kind are encountered in a myriad of contexts, for
instance, to describe large systems of indistinguishable agents such as cell
colonies, flocks of birds, schools of fish, herds of sheep. The finite volume
method we propose is constructed to conserve mass and preserve positivity.
Moreover, convex functionals of the discrete solution are controlled, which we
use to show the convergence of the scheme. Finally, we investigate the scheme
numerically.
