The scalar initial-boundary problem, defined by the first-order hyperbolic partial differential equation u_t + f(x)_x = 0 with (x,t) ∈ R₊ × R₊, is studied. It is assumed that u(0,t) = u_b (t) = u_- when t ∈ R₊ and u(x,0) = u₀(x) = u₊ when x ∈ R₊, where (i) u_- and u₊ are two constants, and (ii) u_- ≠ u₊. Moreover, it is assumed that f ∈ C². Under these conditions, some terms for the global error estimation are derived for two major cases.

The authors start in section 1 with the introduction of several definitions, the most important for the remaining results being the definition of a weak entropy solution of the problem studied. Several lemmas are then formulated and proven in section 2. The main results are presented in section 3, where three theorems are proven. The first two theorems deal with the case of convex fluxes, while the last presents results related to the error estimation in the case of nonconvex fluxes.