Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations

The convergence and accuracy of Adomian's decomposition method of solution is analysed in the context of its application to the solution of Lorenz equations which govern at lower order the convection in a porous layer (or respectively in a pure fluid layer) heated from below. Adomian's decomposition method provides an analytical solution in terms of an infinite power series and is applicable to a much wider range of heat transfer problems. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The analysis indicates that the series converges within a sufficiently small time domain, a result that proves to be significant in the derivation of the practical procedure to compute the infinite power series. Comparison of the results obtained by using Adomian's decomposition method with corresponding results obtained by using a numerical Runge-Kutta-Verner method show that both solutions agree up to 12-13 significant digits at subcritical conditions, and up to 8-9 significant digits at certain supercritical conditions, the critical conditions being associated with the loss of linear stability of the steady convection solution. The difference between the two solutions is presented as projections of trajectories in the state space, producing similar shapes that preserve under scale reduction or magnification, and are presumed to be of a fractal form. (C) 2000 Elsevier Science Ltd. All rights reserved.