  In this paper we consider the numerical solution of a singularly perturbed
one-dimensional semilinear reaction-diffusion problem. A class of differential
schemes is constructed. There is a proof of the existence and uniqueness of the
numerical solution for this constructed class of differential schemes. The
central result of the paper is an $\varepsilon$--uniform convergence of the
second order $\mathcal{O}\left(1/N^2 \right),$ for the discrete approximate
solution on the modified Bakhvalov mesh. At the end of the paper there are
numerical experiments, two representatives of the class of differential schemes
are tested and it is shown the robustness of the method and concurrence of
theoretical and experimental results.
