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

Peter Vadasz, Shmuel Olek

Research output: Contribution to journalArticle

62 Citations (Scopus)

Abstract

The convergence and accuracy of Adomian's decomposition method of solution is analyzed 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.

Original languageEnglish (US)
Pages (from-to)1715-1734
Number of pages20
JournalInternational Journal of Heat and Mass Transfer
Volume43
Issue number10
DOIs
StatePublished - Mar 2000
Externally publishedYes

Fingerprint

Decomposition
decomposition
power series
digits
convection
Runge-Kutta method
Runge Kutta methods
magnification
Fractals
fractals
derivation
projection
heat transfer
Trajectories
trajectories
Heat transfer
Fluids
fluids
approximation
Convection

ASJC Scopus subject areas

  • Fluid Flow and Transfer Processes
  • Energy(all)
  • Mechanical Engineering

Cite this

Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations. / Vadasz, Peter; Olek, Shmuel.

In: International Journal of Heat and Mass Transfer, Vol. 43, No. 10, 03.2000, p. 1715-1734.

Research output: Contribution to journalArticle

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