# Limits of parallelism in explicit ODE methods

- 32 Downloads
- 3 Citations

## Abstract

Numerical methods for ordinary initial value problems that do not depend on special properties of the system are usually found in the class of linear multistage multivalue methods, first formulated by J.C. Butcher. Among these the explicit methods are easiest to implement. For these reasons there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in parallel. Each such group of concurrent function evaluations can be regarded as a stage of the method. However, it turns out that parallelism affords only limited opportunity for reducing the computing time with such methods. This is most evident for the simple linear homogeneous constant-coefficient test problem, whose solution is essentially a matter of approximating the exponential by an algebraic function. For a given number of stages and a given number of saved values, parallelism offers a somewhat enlarged set of algebraic functions from which to choose. However, there is absolutely no benefit in having the degree of parallelism (number of processors) exceed the number of saved values of the method. Thus, in particular, parallel one-step methods offer no speedup over serial one-step methods for the standard linear test problem. Although the implication of this result for general nonlinear problems is unclear, there are indications that dramatic speedups are not possible in general. Also given are some results relevant to the construction of methods.

### Subject classification

AMS(MOS) 65L05 CR G.1.7, G.1.0## Preview

Unable to display preview. Download preview PDF.

### References

- [1]A. Bellen, Parallelism across the steps for difference and differential equations, in:
*Numerical Methods for Ordinary Differential Equations*, eds. A. Bellen, C.W. Gear and E. Russo, Proceedings, L'Aquila, 1987, Lecture Notes in Mathematics no. 1386 (Springer, Berlin, 1989) pp. 22–35.CrossRefGoogle Scholar - [2]L.G. Birta and O. Abou-Rabia, Parallel block predictor-corrector methods for ODE's IEEE Trans. Comput. C-36 (1987) 299–311.CrossRefGoogle Scholar
- [3]J.C. Butcher, On the convergence of numerical solutions to ordinary differential equations, Math. Comp. 20 (1966) 1–10.CrossRefMATHMathSciNetGoogle Scholar
- [4]J.C. Butcher,
*The Numerical Analysis of Ordinary Differential Equations-Runge-Kutta and General Linear Methods*(Wiley, New York, 1987).MATHGoogle Scholar - [5]M.T. Chu and H. Hamilton, Parallel solution of ODE's by multi-block methods, SIAM J. Sci. Statist. Comput. 8 (1987) 342–353.CrossRefMATHMathSciNetGoogle Scholar
- [6]P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev. 27 (1985) 505–536.CrossRefMATHMathSciNetGoogle Scholar
- [7]L. Fox, Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations, Proc. Roy. Soc. London Ser. A 190 (1947) 31–59.CrossRefMATHMathSciNetGoogle Scholar
- [8]C.W. Gear, The potential for parallelism in ordinary differential equations, in:
*Computational Mathematics II*, ed. S.O. Fatunla (Boole Press, Dublin, 1987) pp. 33–48.Google Scholar - [9]C.W. Gear, Parallel methods for ordinary differential equations, Calcolo: Quart. Num. Anal. Theory Comput. 25 (1989) 1–20.CrossRefMathSciNetGoogle Scholar
- [10]E. Hairer, S.P. Nørsett and G. Wanner,
*Solving Ordinary Differential Equations I: Non-Stiff Systems*(Springer, Berlin, 1987).CrossRefGoogle Scholar - [11]S. Horiguchi, Y. Kawazoe and H. Nara, A parallel algorithm for the integration of ordinary differential equations,
*Proc. Int. Conf. on Parallel Processing*(1984) pp. 465–469.Google Scholar - [12]K.R. Jackson and S.P. Norsett, The potential for parallelism in Runge-Kutta methods. Part I: RK formulas in standard form, in preparation.Google Scholar
- [13]W.L. Miranker and W. Liniger, Parallel methods for the numerical integration of ordinary differential equations, Math. Comp. 21 (1967) 303–320.CrossRefMATHMathSciNetGoogle Scholar
- [14]J. Nievergelt, Parallel methods for integrating ordinary differential equations, Comm. ACM 7 (1964) 731–733.CrossRefMATHMathSciNetGoogle Scholar
- [15]L.F. Shampine and H.W. Watts, Block implicit one-step methods, Math. Comp. 23 (1969) 731–740.CrossRefMATHMathSciNetGoogle Scholar
- [16]R.D. Skeel, Waveform iteration and the shifted Picard splitting, SIAM J. Sci. Statist. Comput. 10 (1989) 756–776.CrossRefMATHMathSciNetGoogle Scholar
- [17]R.D. Skeel and H.-W. Tam, Potential for parallelism in explicit linear methods,
*Proc. IMA Conf. on Computational ODEs*, eds. J.R. Cash and I. Gladwell (Oxford Univ. Press), to appear.Google Scholar - [18]H.J. Stetter,
*Analysis of Discretization Methods for Ordinary Differential Equations*(Springer, Berlin, 1973).CrossRefMATHGoogle Scholar - [19]H.-W. Tam, Parallel methods for the numerical solution of ordinary differential equations, Univ. of Illinois at Urbana-Champaign, Report No. UIUCDCS-R-89-1516 (1989); also, Ph.D. thesis.Google Scholar
- [20]H.-W. Tam, Parallel predictor-corrector block methods for ordinary differential equations,
*Proc. 4th SIAM Conf. on Parallel Processing for Scientific Computing*eds. J. Dongarra, P. Messina, D. Sorensen and R. Voigt (SIAM, Philadelphia, 1990) pp. 204–209.Google Scholar - [21]H.-W. Tam, A new parallel algorithm for the numerical solution of ordinary differential equations,
*Proc. IMA Conf. on Computational ODEs*, eds. J.R. Cash and I. Gladwell (Oxford Univ. Press), to appear.Google Scholar - [22]H.-W. Tam, One-stage parallel methods for the numerical solution of ordinary differential equations, SIAM J. Sci. Statist. Comput 13 (1992), to appear.Google Scholar
- [23]H.-W. Tam, Two-stage parallel methods for the numerical solution of ordinary differential equations, SIAM J. Sci. Statist. Comput 13 (1992), to appear.Google Scholar
- [24]H.-W. Tam and R.D. Skeel, Stability of parallel explicit ODE methods, manuscript (1991).Google Scholar
- [25]P.B. Worland, Parallel methods for the numerical solution of ordinary differential equations, IEEE Trans. Comput. C-25 (1976) 1045–1048.CrossRefMATHMathSciNetGoogle Scholar