Compact difference scheme for the differential equation with piecewise-constant coefficient

We present compact difference approximation on equidistant grid for the Dirichlet problem for the second order divergent type linear differential equation with piecewise-constant coefficient and discontinuous right hand side. Our numerical experiments demonstrate essential advantage of the scheme in comparison with the classic divergence scheme. We obtain same results for the Sturm–Liouville problem. The Richardson extrapolation method improves the order of accuracy in both cases.