Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations
type: Item
instance of: scholarly article
title: Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations
zbMATH Open document ID: 1015.76075
DOI: 10.1016/S0017-9310(99)00260-4
author: P. Vadasz
author: Shmuel Olek
published in: International Journal of Heat and Mass Transfer
publication date: 2003-07-17T00:00:00Z
review text: This paper analyses the convergence and accuracy of Adomian's decomposition method applied to the solution of Lorenz equations which govern at lower order the convection in a porous layer (or in a pure fluid layer) heated from below. The authors state that the decomposition method provides an analytical solution in terms of infinite power series. First, the authors formulate the initial value problem for Lorenz equations which consist of three ordinary differential equations in dependent variables \(X(x,y,z;t),\) \( Y(x,y,z;t)\) and \(Z(x,y,z;t),\) which represent the amplitudes of spatial modes for the dimensionless stream function and temperature; the independent variables \(x, y, z\) and \(t\) are the horizontal length, horizontal width, vertical distance and time, respectively. The complete three-dimensional equations give at lower order the solution to the problem of convection in a fluid-saturated porous layer (or in a pure fluid layer) heated from below. The established equations are then solved both analytically using the Adomian's decomposition method and numerically using the Runge-Kutta-Verner method in a wide range of values of the scaled Rayleigh number. It is shown that both solutions agree up to 12-13 significant digits under subcritical conditions, and up to 8-9 significant digits under certain supercritical conditions, the critical conditions being associated with the loss of linear stability of steady convection solution. The results of comparison between computational (Adomian decomposition) and numerical (Runge-Kutta) solutions are presented in 8 figures as projections of trajectories in phase space, producing similar shapes that remain invariant under scale reduction or magnification, and thus are presumed to be of fractal form. Reviewer thinks that this paper is a new step in solving problems which are known to possess properties of sensivity of their solution to small variations in initial conditions.
Mathematics Subject Classification ID: 76R10
Mathematics Subject Classification ID: 76M25
Mathematics Subject Classification ID: 37N10
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 1502289
zbMATH Keywords: fractal solution
zbMATH Keywords: free convection
zbMATH Keywords: chaos
zbMATH Keywords: weak turbulence
zbMATH Keywords: convergence
zbMATH Keywords: accuracy
zbMATH Keywords: Adomian's decomposition method
zbMATH Keywords: Lorenz equation
zbMATH Keywords: pure fluid layer
zbMATH Keywords: analytical solution
zbMATH Keywords: infinite power series
zbMATH Keywords: fluid-saturated porous layer
zbMATH Keywords: Runge-Kutta-Verner method
reviewed by: Ioan Pop
MaRDI profile type: Publication
Wikidata QID: Q127564961
full work available at URL: s0017-9310(99)00260-4
OpenAlex ID: W1994347159
description: scientific article; zbMATH DE number 1502289