Lo - stable methods for parabolic partial differential equations

Abstract

In recent years much attention has been devoted in the literature to the development, analysis and implementation of extrapolation methods for the numerical solution of partial differential equations with mixed initial and boundary values specified, see, for example, Lawson and Morris [5], Lawson and Swayne [6] and Gourlay and Morris [3]. The essential theme of these papers was to develop Lo-stable methods for the solution of parabolic partial differential equations in which splitting methods, such as the Crank-Nicolson method, are less than satisfactory when a time discretization is used with time steps which are too large relative to the spatial discretization. In the present paper a family of new Lo-stable methods based on Padé approximants to the exponential function is developed, and. higher accuracy is achieved. The methods are tested on heat equations in one and two space dimensions in which discontinuities exist between the initial and boundary conditions.