An analysis of conservative finite difference schemes for differential equations with discontinuous coefficients

A class of monotone conservative schemes is derived for the boundary value problem related to the Sturm-Liouville operator Au≔-(k(x)u′(x))′+q(x)u(x), with discontinuous coefficient k = k(x). The discrete analogous of the law of conservation are compared for the finite element and finite difference approaches. In the class of discontinuous coefficients, the necessary condition for conservativeness of the finite difference scheme is derived. The obtained one parametric family of conservative schemes permits one to construct new conservative schemes. The examples, presented for different discontinuous coefficients, and results show how the conservativeness conditions need to be taken into account in numerical solving boundary value problems.