  A class of different schemes for the numerical solving of semilinear
singularly--perturbed reaction--diffusion boundary--value problems was
constructed. The stability of the difference schemes was proved, and the
existence and uniqueness of a numerical solution were shown. After that, the
uniform convergence with respect to a perturbation parameter $\varepsilon$ on a
modified Shishkin mesh of order 2 has been proven. For such a discrete
solution, a global solution based on a linear spline was constructed, also the
error of this solution is in expected boundaries. Numerical experiments at the
end of the paper, confirm the theoretical results. The global solutions based
on a natural cubic spline, and the experiments with Liseikin, Shishkin and
modified Bakhvalov meshes are included in the numerical experiments as well.
