Part 1: Singular Perturbation

Abstract

We can expect any limiting solution Uo(t) to satisfy the reduced equation f(Uo)Uo + g(Uo) = 0, away from boundary and interior layer regions of rapid change, together with appropriate endpoint and jump conditions. Very successful numerical methods have been developed for solving examples with specific functions f and g and specific values A, B, and T. They are generally based on upwinding the finite-difference approximation to u by taking the sign of f into account (cf., e.g., Osher [20) and Lorenz [13),[14)). Associated theories imply the existence of a unique solution u(t,E) provided g(O) = 0 and g'( u) remains nonpositive. Most impressive, these methods are able to compute limiting solutions regardless of how many "turning points," where f( u) = 0, are encountered. Other reliable computing procedures determine u as the steady state of the corresponding parabolic problem.