Analysis of the upwind finite volume method for general initial- and boundary-value transport problems

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initialand boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L^∞ and the advection vector field υ has low regularity properties, namely υ ∈ L¹(]0, T [, (W^1,1(Ω))d), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in L^p(Ω), with p < +∞, of the approximate solution towards the unique weak solution of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L¹. This is nothing but the discrete counterpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up to now.