Comparisons of numerical solution methods for differential equations with discontinuous coefficients

There are many important physical systems that can be modeled using differential equations with discontinuous coefficients. If such problems are approximated numerically, then the usual analysis of accuracy fails because of the discontinuities. Eight different approximations to a one-dimensional steady-state boundary-value problem for a general symmetric second-order ordinary differential equation with discontinuous leading coefficient are studied in this paper. Two modeling situations with a single discontinuity are considered: (1) the discontinuity can be located accurately relative to the grid spacing; and (2) the discontinuity is at some random point in a grid interval.