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Parallel defect control

  • Part II Numerical Mathematics
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Abstract

How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p − 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.

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References

  1. Bryant, P. J.:Nonlinear wave groups in deep water, Studies in Applied Mathematics (1979), 1–30.

  2. Burrage, K.:Solving nonstiff IVPs in a transputer environment, Manuscript (1990), CMSR, University of Liverpool, U.K.

    Google Scholar 

  3. Cash, J. R.:A block 6(4) Runge-Kutta formula for nonstiff initial value problems, ACM Trans. Math. Soft. 15 (1989), 15–28.

    Google Scholar 

  4. Davis, P. J.:Interpolation and Approximation, Dover, New York, 1975.

    Google Scholar 

  5. Dormand, J. R. and Prince, P. J.:Runge-Kutta triples, Comp. and Maths. with Appls. 12 (1986), 1007–1017.

    Google Scholar 

  6. Enright, W. H.:A new error-control for initial value solvers, Applied Maths. and Computation 31 (1989), 288–301.

    Google Scholar 

  7. Enright, W. H.:Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal. 26 (1989), 588–599.

    Google Scholar 

  8. Enright, W. H., Jackson, K. R., Nørsett, S. P. and Thomsen, P. G.:Interpolants for Runge-Kutta formulas, ACM Trans. Math. Soft. 12 (1986), 193–218.

    MathSciNet  Google Scholar 

  9. Enright, W. H. and Pryce, J. D.:Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Soft. 13 (1987), 1–27.

    Google Scholar 

  10. Enright, W. H. and Higham, D. J.:Parallel defect control, Technical Report No. 237/90, Department of Computer Science, University of Toronto, Canada (1990).

    Google Scholar 

  11. Gladwell, I.:Initial value routines in the NAG library, ACM Trans. Math. Soft. 5 (1979), 386–400.

    Google Scholar 

  12. Gladwell, I., Shampine, L. F., Baca, L. S. and Brankin, R. W.:Practical aspects of interpolation in Runge-Kutta codes, SIAM J. Sci. Stat. Comput. 8 (1987), 322–341.

    Google Scholar 

  13. Hairer, E., Nørsett, S. P. and Wanner, G.:Solving Ordinary Differential Equations I, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  14. Higham, D. J.:Robust defect control with Runge-Kutta schemes, SIAM J. Numer. Anal. 26 (1989), 1175–1183.

    Google Scholar 

  15. Higham, D. J.:Highly continuous Runge-Kutta interpolants, Technical Report No. 220/89, Department of Computer Science, University of Toronto, Canada (1989); to appear in ACM Trans. Math. Softw.

    Google Scholar 

  16. Higham, D. J.:Runge-Kutta defect control using Hermite-Birkhoff interpolation, Technical Report No. 221/89, Department of Computer Science, University of Toronto (1989); to appear in SIAM J. Sci. Stat. Comput.

  17. Higham, D. J.:Global error versus tolerance for explicit Runge-Kutta methods, Technical Report No. 233/90, Department of Computer Science, University of Toronto, Canada (1990); to appear in IMA J. Numer. Anal.

    Google Scholar 

  18. Houwen, P. J. van der and Sommeijer, B. P.:Variable step iteration of high-order Runge-Kutta methods on parallel computers, Report NM-R8817, Centre for Mathematics and Computer science, Amsterdam, 1988.

    Google Scholar 

  19. Houwen, P. J. van der and Sommeijer, B.P.:Block Runge-Kutta methods on parallel computers, Report NM-R8906, Centre for Mathematics and Computer science, Amsterdam, 1989.

    Google Scholar 

  20. Jackson, K. R. and Nørsett, S. P.:The potential for parallelism in Runge-Kutta methods. Part 1:RK formulas in standard form, Technical Report No. 239/90, Department of Computer Science, University of Toronto, Canada (1990).

    Google Scholar 

  21. Owren, B. and Zennaro, M.:Derivation of efficient continuous explicit Runge-Kutta methods, Technical Report No. 240/90, Department of Computer Science, University of Toronto, Canada (1990).

    Google Scholar 

  22. Shampine, L. F.:Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), 1014–1027.

    Google Scholar 

  23. Shampine, L. F.:Some practical Runge-Kutta formulas, Math. Comp. 46 (1986), 135–150.

    Google Scholar 

  24. Sharp, P. W. and Smart, E.: Private communication, (1990).

  25. Skeel, R. D. and Tam, H.-W.:Potential for parallelism in explicit linear methods, Working Document, Department of Computer Science, University of Illinois at Urbana-Champaign (1989).

  26. Stetter, H. J.:Considerations concerning a theory for ODE-solvers, inNumerical Treatment of Differential Equations: Proc. Oberwolfach, 1976 (Eds. R. Bulirsch, R. D. Grigorieff and J. Schröder), Lecture Notes in Mathematics 631, Springer, Berlin, (1978) 188–200.

    Google Scholar 

  27. Stetter, H. J.:Global error estimation in Adams PC-codes, ACM Trans. Math. Soft. 5 (1979), 415–430.

    Google Scholar 

  28. Stewart, N. F.:Certain equivalent requirements of approximate solutions of x′(t, x), SIAM J. Numer. Anal. 7 (1970), 256–270.

    Google Scholar 

  29. Tam, H.-W.:Parallel methods for the numerical solution of ordinary differential equations, Report No. UIUCDCS-R-89-1516, Department of Computer Science, University of Illinois at Urbana-Champaign (1989).

  30. Verner, J. H.:On the derivation of higher order interpolants, Seminar presented at “The 1990 Conference on the Numerical Solution of ODEs”, Helsinki, Finland, June 1990.

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Both authors were supported by the Information Technology Research Centre of Ontario, and the Natural Sciences and Engineering Research Council of Canada. Second author's current address: Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland.

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Enright, W.H., Higham, D.J. Parallel defect control. BIT 31, 647–663 (1991). https://doi.org/10.1007/BF01933179

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