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Convergence to the entropy solution of a second-order scheme for the resolution of scalar hyperbolic conservation laws is studied. We consider the case of the so-called two-point upstream weighting scheme, widely used in petroleum engineering, to solve fluid flows in porous media problems. We prove convergence of the two-point upstream weighting scheme to the entropy solution for both discrete and semidiscrete approximations. with initial condition u(x, 0) = u0(x). We assume f £ C and smoothdoi:10.1090/s0025-5718-1991-1094962-2 fatcat:23bzmx5bwbdzjh4wzxg5lkigpq