Using a spline in tension for the problem: $-\varepsilon y''+p(x)y=f(x)$, $0<x<1$, $y(0)=\alpha_0$, $y(1)=\alpha_1$; $0<\varepsilon \ll1$, a family of difference schemes is derived. The schemes have a second order of uniform convergence. Some of them converge with respect to $\varepsilon$.