The use of nonrational functions to represent steep front solutions to partial differential equations

Abstract

It is shown, by calculated examples, that it is possible to follow successfully the development and evolution of steep fronts (almost to but not including actual discontinuities) in the solution of time-dependent partial differential equations by means of global non-rational representations. A characteristic steplike shape forms the basis for the representations; the hyperbolic tangent and a function involving a square root, both having this characteristic shape, gave similar results. The Method of Lines, specifically as formulated by Gelinas, Doss and Miller ( J. Comput. Phys. 40 (1981), 202–249) was used to form a system of ordinary differential equations governing the time-variation of the parameters in the representation. A fixed-shape moving solution of the wave equation, and developing steep fronts in solutions of Burgers' equation and of the Buckley-Leverett equation were successfully computed. The results are all displayed in the form of plotted profiles, and are compared with the corresponding results in the literature.