Skip to main content
SpringerLink
Log in

A novel series of solution methods for solving nonlinear stiff dynamic problems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A novel type of linear multi-step formulas is proposed for solving initial value problems, such as the problems of multi-body systems and vibration systems and a variety of dynamic problems in engineering. This type of formulas is derived from an eigen-based theory and is characterized by problem dependency since it has problem-dependent coefficients, which are functions of physical properties for defining the problem under analysis and applied step size. These coefficients are no longer limited to be scalar constants but can be constant matrices. The detailed development of a set of problem-dependent formulas is presented, and their numerical properties are intensively explored. These formulas can be explicitly or implicitly implemented in the solution of initial value problems. One of these problem-dependent methods can combine A-stability and explicit formulation together, and thus, it is best suited to solve nonlinear stiff problems, such as problems in chemical kinetics and vibrations, the analysis of control systems and the study of dynamical systems. It is validated that A-stability in conjunction with explicit formulation can save many computational efforts for solving nonlinear stiff dynamic problems when compared to conventional implicit methods. Notice that there exists no conventional method that can simultaneously combine A-stability and explicit formulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. John Wiley, New York (1962)

    MATH  Google Scholar 

  2. Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Natl. Acad. Sci. USA 38(3), 235–243 (1952)

    Article  MathSciNet  Google Scholar 

  3. Liniger, W., Willoughby, R.A.: Efficient integration methods for stiff systems of ordinary differential equations. SIAM J. Numer. Anal. 7(1), 47–66 (1970)

    Article  MathSciNet  Google Scholar 

  4. Donelson, J., Hansen, E.: Cyclic composite multistep predictor-corrector methods. SIAM J. Numer. Anal. 8(1), 137–157 (1971)

    Article  MathSciNet  Google Scholar 

  5. Gupta, G.K., Wallace, C.S.: Some new multistep methods for solving ordinary differential equations. Math. Comput. 29, 489–500 (1975)

    Article  MathSciNet  Google Scholar 

  6. Cash, J.R.: On a class of cyclic methods for the numerical integration of stiff systems of ODEs. BIT 17, 270–280 (1977)

    Article  MathSciNet  Google Scholar 

  7. Bui, T.D., Bui, T.R.: Numerical methods for extremely stiff systems of ordinary differential equations. Appl. Math. Model. 3(5), 355–358 (1979)

    Article  Google Scholar 

  8. Petzold, L.R.: An efficient numerical method for highly oscillatory ordinary differential equations. SIAM J. Numer. Anal. 18(3), 455–479 (1981)

    Article  MathSciNet  Google Scholar 

  9. Chang, S.Y.: A new family of explicit method for linear structural dynamics. Comput. Struct. 88(11–12), 755–772 (2010)

    Article  Google Scholar 

  10. Hussein, B.A., Shabana, A.A.: Sparse matrix implicit numerical integration of the stiff differential/algebraic equations: Implementation. Nonlinear Dyn. 65, 369–382 (2011)

    Article  MathSciNet  Google Scholar 

  11. Chang, S.Y.: Dissipative, non-iterative integration algorithms with unconditional stability for mildly nonlinear structural dynamics. Nonlinear Dyn. 79(2), 1625–1649 (2015)

    Article  Google Scholar 

  12. Guillot, L., Cochelin, B., Vergez, C.: A Taylor series-based continuation method for solutions of dynamical systems. Nonlinear Dyn. 98, 2827–2845 (2019)

    Article  Google Scholar 

  13. Chang, S.Y.: A dual family of dissipative structure-dependent integration methods. Nonlinear Dyn. 98(1), 703–734 (2019)

    Article  Google Scholar 

  14. Chang, S.Y.: Non-iterative methods for dynamic analysis of nonlinear velocity-dependent problems. Nonlinear Dyn. 101, 1473–1500 (2020)

    Article  Google Scholar 

  15. Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956)

    Article  MathSciNet  Google Scholar 

  16. Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)

    Article  MathSciNet  Google Scholar 

  17. Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice- Hall, Englewood Cliffs, NJ (1971)

    MATH  Google Scholar 

  18. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. John Wiley, New York (1973)

    MATH  Google Scholar 

  19. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley, Chichester, UK (1987)

    MATH  Google Scholar 

  20. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Spring-Verlag, Berlin, Heidelberg (1987)

    Book  Google Scholar 

  21. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The initial value Problem. John Wiley, Chichester, UK (1991)

    MATH  Google Scholar 

  22. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Spring-Verlag, New York (1993)

    MATH  Google Scholar 

  23. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  24. Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)

    Article  MathSciNet  Google Scholar 

  25. Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974)

    Article  MathSciNet  Google Scholar 

  26. Spijker, M.N.: Step size restrictions for stability of one-step methods in the numerical solution of initial value problems. Math. Comput. 45, 377–392 (1985)

    Article  MathSciNet  Google Scholar 

  27. Majda, G.: Stability with large step sizes for multistep discretizations of stiff ordinary differential equations. Math. Comput. 47, 473–502 (1986)

    Article  MathSciNet  Google Scholar 

  28. Chang, S.Y.: Explicit pseudodynamic algorithm with unconditional stability. J. Eng. Mech. ASCE 128(9), 935–947 (2002)

    Article  Google Scholar 

  29. Dieci, L., Estep, D.: Some stability aspects of schemes for the adaptive integration of stiff initial value problems. SIAM J. Sci. Stat. Comput. 12(6), 1284–1303 (1991)

    Article  MathSciNet  Google Scholar 

  30. E.J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, Macmillan, 1877.

  31. Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. (Later Wirel. World) 1, 701–710 (1920)

    Google Scholar 

  32. Van der Pol, B.: On relaxation-oscillations, The London, Edinburgh and Dublin Phil. Mag. J. Sci. 2(7), 978–992 (1926)

    Google Scholar 

Download references

Acknowledgements

The author is grateful to acknowledge that this study is financially supported by the Ministry of Science and Technology, Taiwan, R.O.C., under Grant No. MOST-109-2221-E-027-002.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Taipei University of Technology, NTUT, No. 1, Section 3, Jungshiau East Road, Box 2653, Taipei, 10608, Taiwan, Republic of China

    Shuenn-Yih Chang

Authors
  1. Shuenn-Yih Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuenn-Yih Chang.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, SY. A novel series of solution methods for solving nonlinear stiff dynamic problems. Nonlinear Dyn 107, 2539–2562 (2022). https://doi.org/10.1007/s11071-021-07048-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07048-0

Keywords

Search

Navigation