The numerical solution of non-linear partial differential equations by the method of lines

Abstract

A new method for solving non-linear parabolic partial differential equations, based on the method of lines, is developed. Second order and fourth order finite difference approximations to the spatial derivatives are used, and in each case the band structure of the associated Jacobian is exploited. This, together with the use of Gear's fourth order stiffly stable method in Nordsieck form, leads to a method which compares favourably with the respected Sincovec and Madsen method on Burgers' equation. The method has been tested on a number of difficult problems in the literature and has proved to be most successful.