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. In case (1), all methods produce the exact solution if the discontinuity is properly located in the grid. Moreover, the error in the methods is proportional to the error in the location of the discontinuity. Consequently those methods that most easily take advantage of the location of the discontinuity are better in this situation. In case (2), all methods are first-order accurate and essentially of the same quality. The first-order accuracy also applies to the fluxes. These algorithms are being analyzed in preparation for applying them to porous-media flows where the fluxes are used for contaminant tracking, so the accuracy of the fluxes is important. Also, a novel justification for harmonic averaging is given. Two-dimensional extensions are in progress.