Skip to main content
SpringerLink
Log in

Parallel predictor-corrector iteration of pseudo two-step RK methods for nonstiff IVPs

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

A parallel predictor-corrector (PC) iteration scheme for a general class of pseudo two-step Runge-Kutta methods (PTRK methods) of arbitrarily high order is analyzed for solving first-order nonstiff initial-value problems (IVPs) on parallel computers. Starting with ans-stage pseudo two-step RK method of orderp * withw implicit stages, we apply the highly parallel PC iteration process in P(EC)mE mode. The resulting parallel-iterated pseudo two-step RK method (PIPTRK method) uses an optimal number of processors equal tow. By a number of numerical experiments, we show the superiority of the PIPTRK methods proposed in this paper over both sequential and parallel methods available in the literature.

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

Similar content being viewed by others

References

  1. K. Burrage, Efficient block predictor-corrector methods with a small number of corrections. J. Comput. Appl. Math.,45 (1993), 139–150.

    Article  MATH  MathSciNet  Google Scholar 

  2. K. Burrage, Parallel methods for initial value problems. Appl. Numer. Math.,11 (1993), 5–25.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations. Clarendon Press, Oxford, 1995.

    MATH  Google Scholar 

  4. K. Burrage and H. Suhartanto, Parallel iterated methods based on multistep Runge-Kutta mehods of Radau type. Advances in Computational Mathematics,7 (1997), 37–57.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.C. Butcher, The Numerial Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Wiley, New York, 1987.

    Google Scholar 

  6. M.T. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods. SIAM J. Sci. Statist. Comput.,3 (1987), 342–353.

    MathSciNet  Google Scholar 

  7. N.H. Cong, Parallel iteration of symmetric Runge-Kutta for nonstiff initial-value problems. J. Comput. Appl. Math.,51 (1994), 117–125.

    Article  MATH  MathSciNet  Google Scholar 

  8. N.H. Cong, Explicit pseudo two-step Runge-Kutta methods for parallel computers. Intern. J. Comput. Math.,73 (1999), 77–91.

    Article  MATH  MathSciNet  Google Scholar 

  9. N.H. Cong, Continuous variable stepsize explicit pseudo two-step RK methods. J. Comput. Appl. Math.,101 (1999), 105–116.

    Article  MATH  MathSciNet  Google Scholar 

  10. N.H. Cong, A general family of pseudo two-step Runge-Kutta methods. Southeast Asia Bull. Math.,25 (2001), 61–73.

    Article  MATH  MathSciNet  Google Scholar 

  11. N.H. Cong and T. Mitsui, Collocation-based two-step Runge-Kutta methods. Japan J. Indust. Appl. Math.,13 (1996), 171–183.

    Article  MATH  MathSciNet  Google Scholar 

  12. N.H. Cong and T. Mitsui, A class of explicit parallel two-step Runge-Kutta methods. Japan J. Indust. Appl. Math.,14 (1997), 303–313.

    Article  MathSciNet  Google Scholar 

  13. N.H. Cong, H. Podhaisky and R. Weiner, Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer. Comput. Math. Appl.,36 (1998), 107–116.

    Article  MATH  MathSciNet  Google Scholar 

  14. N.H. Cong and H.T. Vi, An improvement for explicit parallel Runge-Kutta methods. Vietnam J. Math.,23 (1995), 241–252.

    MATH  MathSciNet  Google Scholar 

  15. A.R. Curtis, High-order explicit Runge-Kutta formulae, their uses and limitations. J. Inst. Math. Appl.,16 (1975), 35–55.

    Article  MATH  MathSciNet  Google Scholar 

  16. A.R. Curtis, Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. Her Majesty’s Stationary Office, London, 1964.

    MATH  Google Scholar 

  17. E. Hairer, A Runge-Kutta method of order 10. J. Inst. Math. Appl.,21 (1978), 47–59.

    Article  MathSciNet  Google Scholar 

  18. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations, I. Nonstiff Problems (2nd edition). Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  19. P.J. van der Houwen and N.H. Cong, Parallel block predictor-corrector methods of Runge-Kutta type. Appl. Numer. Math.,13 (1993), 109–123.

    Article  MATH  MathSciNet  Google Scholar 

  20. P.J. van der Houwen and B.P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control. J. Comput. Appl. Math.,29 (1990), 111–127.

    Article  MATH  MathSciNet  Google Scholar 

  21. P.J. van der Houwen and B.P. Sommeijer, Block Runge-Kutta methods on parallel computers. Z. Angew. Math. Mech.,68 (1992), 3–10.

    Google Scholar 

  22. T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal.,9 (1972), 603–637.

    Article  MATH  MathSciNet  Google Scholar 

  23. S.P. Nørsett and H.H. Simonsen, Aspects of parallel Runge-Kutta methods. Numerical Methods for Ordinary Differential Equations (eds. A. Bellen, C. W. Gear and E. Russo), Proceedings L’Aquilla 1987, Lecture Notes in Mathematics1386, Springer-Verlag, Berlin, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Nguyen Huu Cong

  2. Graduate School of Human Informatics, Nagoya University, Puro-cho, Chikusa-ku, 464-8601, Nagoya, Japan

    Taketomo Mitsui

Authors
  1. Nguyen Huu Cong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taketomo Mitsui
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by Scientist Exchange Program FY2001 of JSPS.

About this article

Cite this article

Huu Cong, N., Mitsui, T. Parallel predictor-corrector iteration of pseudo two-step RK methods for nonstiff IVPs. Japan J. Indust. Appl. Math. 20, 51 (2003). https://doi.org/10.1007/BF03167462

Download citation

  • Received:

  • Accepted:

  • DOI: https://doi.org/10.1007/BF03167462

Key words

Search

Navigation