Open Access

ARTICLE

A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations

by Chein-Shan Liu¹, Satya N. Atluri²

1 Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw
2 Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2008, 31(2), 71-84. https://doi.org/10.3970/cmes.2008.031.071

Abstract

Iterative algorithms for solving a nonlinear system of algebraic equations of the type: F_i(x_j) = 0, i,j = 1,…,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one due to its easy numerical implementation. However, this type of algorithm is sensitive to the initial guess of the solution and is expensive in the computations of the Jacobian matrix ∂ F_i/ ∂ x_j and its inverse at each iterative step. In a time-integration of a system of nonlinear Ordinary Differential Equations (ODEs) of the type B_ijx_j + F_i = 0 where B_ij are nonlinear functions of x_j the methods which involve an inverse of the Jacobain matrix B_ij = ∂ F_i/ ∂ x_j are called "Implicit'', while those that do not involve an inverse of ∂ F_i/ ∂ x_j are called "Explicit''. In this paper a natural system of explicit ODEs is derived from the given system of nonlinear algebraic equations (NAEs), by introducing a fictitious time, such that it is a mathematically equivalent system in the n+1-dimensional space as the original algebraic equations system is in the n-dimensional space. The iterative equations are obtained by applying numerical integrations on the resultant ODEs, which do not need the information of ∂ F_i/ ∂ x_j and its inverse. The computational cost is thus greatly reduced. Numerical examples given confirm that this fictitious time integration method (FTIM) is highly efficient to find the true solutions with residual errors being much smaller. Also, the FTIM is used to study the attracting sets of fixed points, when multiple roots exist.

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