  We develop a semi-discretization approximation for scalar conservation laws
with multiple rough time dependence in inhomogeneous fluxes. The method is
based on Brenier's transport-collapse algorithm and uses characteristics
defined in the setting of rough paths. We prove strong $L^1$-convergence for
inhomogeneous fluxes and provide a rate of convergence for homogeneous one's.
The approximation scheme as well as the proofs are based on the recently
developed theory of pathwise entropy solutions and uses the kinetic formulation
which allows to define globally the (rough) characteristics.
