We deal with initial-boundary value problems for systems of conservation laws in one space dimension and we focus on the boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. In this paper, we establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that, under reasonable assumptions, the self-similar second-order approximation and the classical viscous approximation provide the same limit. Our analysis applies to both the characteristic and the non characteristic case. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields.