A novel time integration method for solving a large system of non-linear algebraic equations

CMES - Computer Modeling in Engineering and Sciences

Volumn 31, Issue 2, 2008, Pages 71-83

  Liu, Chein Shan ^a   Atluri, Satya N ^b  

^a NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Fictitious time integration method (FTIM); Iterative method; Nonlinear algebraic equations; Ordinary differential equations

Indexed keywords

ALGEBRAIC EQUATIONS; ATTRACTING SETS; COMPUTATIONAL COSTS; DATE BACKS; DIMENSIONAL SPACES; EQUIVALENT SYSTEMS; FICTITIOUS TIME INTEGRATION METHOD (FTIM); FIXED POINTS; HIGHLY EFFICIENT; INITIAL GUESSES; ITERATIVE ALGORITHMS; ITERATIVE EQUATIONS; LARGE SYSTEMS; LINEAR ALGEBRAIC EQUATIONS; MULTIPLE ROOTS; NATURAL SYSTEMS; NONLINEAR ALGEBRAIC EQUATIONS; NONLINEAR FUNCTIONS; NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS; NUMERICAL EXAMPLES; NUMERICAL IMPLEMENTATIONS; NUMERICAL INTEGRATIONS; RESIDUAL ERRORS; TIME INTEGRATIONS;

INFORMATION RETRIEVAL SYSTEMS; INTEGRATION; ITERATIVE METHODS; LINEAR EQUATIONS; NONLINEAR SYSTEMS; NUMERICAL ANALYSIS; ORDINARY DIFFERENTIAL EQUATIONS; SOLUTIONS;

NONLINEAR EQUATIONS;

EID: 55349116390     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. Allgower, E. L.; Georg, K. (1990): Numerical Continuation Methods: an Introduction. Springer, New York.
2. Atluri, S. N. (2002): Methods of Computer Modeling in Engineering and Sciences. Tech. Science Press, 1400 pages.
3. Atluri, S. N.; Liu, H. T.; Han, Z. D. (2006): Meshless local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems. CMES: Computer Modeling in Engineering & Sciences, vol. 14, pp. 141-152.
4. Atluri, S. N.; Shen, S. (2002): The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite and boundary element methods. CMES: Computer Modeling in Engineering & Sciences, vol. 3, pp. 11-51.
5. Atluri, S. N.; Zhu, T. L. (1998a): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comp. Mech., vol. 22, pp. 117-127.
6. Atluri, S. N.; Zhu, T. L. (1998b): A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Comp. Model. Simul. Eng., vol. 3, pp. 187-196.
7. Broyden, C. G. (1965): A class of methods for solving nonlinear simultaneous equations. Math. Comp., vol. 19, pp. 577-593.
8. Chow, S. N.; Mallet-Paret, J.; Yorke, J. A. (1978): Finging zeroes of maps: homotopy methods that are constructive with probability one. Math. Comp., vol. 32, pp. 887-889.
9. Davidenko, D. (1953): On a new method of numerically integrating a system of nonlinear equations. Doklady Akad. Nauk SSSR, vol. 88, pp. 601-604.
10. Dennis, J. E. (1971): On the convergence of Broyden's method for nonlinear systems of equations. A class of methods for solving nonlinear simultaneous equations. Math. Comp., vol. 25, pp. 559-567.
11. Dennis, J. E.; More, J. J. (1974): A characterization of superlinear convergence and its application to quasi-Newton method. Math. Comp., vol. 28, pp. 549-560.
12. Dennis, J. E.; More, J. J. (1977): Quasi-Newton methods, motivation and theory. SIAM Rev., vol. 19, pp. 46-89.
13. Deuflhard, P. (2004): Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, New York.
14. Hirsch, M.; Smale, S. (1979): On algorithms for solving f(x) = 0. Commun. Pure Appl. Math., vol. 32, pp. 281-312.
15. Hsu, S. B. (1988): The Numerical Methods for Nonlinear Simultaneous Equations. Central Book Publisher, Taipei, Taiwan.
16. Kellogg, R. B. T.; Li, T. Y.; Yorke, J. A. (1976): A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM J. Num. Anal., vol. 13, pp. 473-483.
17. Li, T. Y.; Yorke, J. A. (1980): A simple reliable numerical algorithm for following homotopy paths. In Analysis and Computation of Fixed Points, Robinson, S. M. ed., pp. 73-91, Academic Press, New York.
18. Li, T. Y. (1997): Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica, vol. 6, pp. 399-436.
19. Liu, C.-S. (2001): Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Meck., vol. 36, pp. 1047-1068.
20. Liu, C.-S. (2006): The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Computer Modeling in Engineering & Sciences, vol. 13, pp. 149-163.
21. Liu, C.-S. (2007a): A modified Trefftz method for two-dimensional Laplace equation considering the domain's characteristic length. CMES: Computer Modeling in Engineering & Sciences, vol. 21, pp. 53-66.
22. Liu, C.-S. (2007b): A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain. CMES: Computer Modeling in Engineering & Sciences, vol. 20, pp. 111-122.
23. Liu, C.-S. (2007c): An effectively modified direct Trefftz method for 2D potential problems considering the domain's characteristic length. Eng. Anal. Bound. Elem., vol. 31, pp. 983-993.
24. Liu, C.-S. (2008a): A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly-connected domains. Numer. Meth. Partial Diff. Eq., vol. 24, pp. 179-192.
25. Liu, C.-S. (2008b): Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Problems, vol. 2008, Article ID 749865.
26. Liu, C.-S. (2008c): Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. J. Sound Vib., vol. 318, pp. 148-165.
27. Liu, C.-S. (2008d): A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. CMES: Computer Modeling in Engineering & Sciences, vol. 27, pp. 137-149.
28. Liu, C.-S.; Chang, J. R.; Chang, K. H.; Chen, Y. W. (2008): Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data. CMC: Computers, Materials & Continua, vol. 7, pp. 97-107.
29. Roose, A.; Kulla, V.; Lomb, M.; Meressoo, T. (1990): Test examples of systems of non-linear equations. Estonian Software and Computer Service Company, Tallin 1990.
30. Spedicato, E.; Hunag, Z. (1997): Numerical experience with Newton-like methods for nonlinear algebraic systems. Computing, vol. 58, pp. 69-89.
31. Zhu, T.; Zhang, J.; Atluri, S. N. (1998): A meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Comp. Mech., vol. 22, pp. 174-186.
32. Zhu, T.; Zhang, J.; Atluri, S. N. (1999): A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem., vol. 23, pp. 375-389.