Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary.

This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in $L^p_{loc}$ for every $p\in [ 1, +\infty)$. This also yields a new proof of the existence of an entropy weak solution.