This paper deals with the solution of the boundary value problem \(\varepsilon y'' + s(t)y' + c(t)y = f(t)\), \(t \in [t_{\min}, t_{\max}]\), \(y(t_{\min}) = a\), \(y(t_{\max}) = b\), where \(\varepsilon\) is positive (it could be very small), \(c(t)\) and \(f(t)\) are continuous functions on the integration interval \([t_{\min}, t_{\max}]\), and \(s(t)\) is differentiable. The presented method is based on second order difference schemes where the convergence is not very fast. It applies a mesh selection strategy derived by using sufficient conditions which ensure the well-conditioning of the corresponding tridiagonal matrices. It has the positive feature of choosing the initial mesh very efficiently and this allows it to perform better than higher order methods, especially for small values of \(\varepsilon\) or for not very high tolerance. Numerical tests are reported.