The authors consider an unsteady one-dimensional hyperbolic equation with a convex scalar flux, such as the Burgers equation. A specific explicit discretization is used with a modified Lax-Friedrichs flux with a smoothing coefficient which varies with the grid resolution \(h\) to increase the number of grid points across any discontinuity as \(h\to 0\). Since the numerical discretization is monotone, for sufficiently small timesteps, the classical results of M. G. Crandall and A. Majda [Math. Comput. 34, 1–21 (1980; Zbl 0423.65052)], and the more recent results of H. Nessyahu and E. Tadmor [SIAM J. Numer. Anal. 29, No. 6, 1505–1519 (1992; Zbl 0765.65092)], prove convergence of the nonlinear discretization to the unique entropy solution. It is proved that for initial data which is smooth apart from one or more discontinuities the corresponding linearized discretization yields solutions which converge pointwise to the analytic solution everywhere except along the discontinuities. It is also proved that the discrete approximation of the linearized perturbation to output integrals converges to the analytic value. It is considered that the correct formulation and discretization of adjoint equations in the presence of shocks is important and that it is the main motivation for the present analysis.