zbMATH — the first resource for mathematics

A parallel shooting technique for solving dissipative ODE’s. (English) Zbl 0788.65079
A new approach is presented designed for a restricted class of ordinary differential equations (ODE’s), where the righ-hand side function is dissipative. A fixed point problem arising from the application of a step-by-step method to a system of ODE’s is adopted. The authors’ formulation is based on the exact solution of the differential system. This new formulation is used to determine what kind of initial value problems should be considered.
A very simple algorithm aimed at solving the fixed point problem is introduced. Its convergence is studied in terms of the logarithmic norm of the right-hand-side. This algorithm already emphasizes the necessity of dealing only with a restricted class of problems. Newton’s method is considered. The class of dissipative problems is shown to be particulary appropriate. Numerical experiments aimed at verifying the theoretical convergence results for the proposed algorithm are presented.
The discrete-time version of the algorithm is studied. Especially the influence of the perturbations arising from the introduction of approximations is analyzed. Simulations are presented that prove of the proposed technique is competitive in situations, where it is not possible to parallelize “across the system”.
Reviewer: I.Dimov (Sofia)

65L05 Numerical methods for initial value problems involving ordinary differential equations
65Y05 Parallel numerical computation
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI
[1] Bellen, A., Vermiglio, R., Zennaro, M.: Parallel ODE-solvers with step-size control. J. Comp. Appl. Math.31, 277–293 (1990). · Zbl 0707.65051
[2] Bellen, A., Zennaro, M.: Parallel algorithms for initial value problems. J. Comp. Appl. Math.25, 341–350 (1989). · Zbl 0675.65134
[3] Birta, L., Abou-Rabia, O.: Parallel block predictor-corrector methods for ODE’s. IEEE Transactions on ComputersC-36, 299–311 (1987).
[4] Chartier, P.: Application of Bellen’s method to ODE’s with dissipative right-hand side. Research Report 593, IRISA, Campus de Beaulieu, Rennes, France, 1991.
[5] Chartier, P.: L-stable parallel one-block methods for ordinary differential equations. SIAM J. Numer. Anal. (1993) (to appear). · Zbl 0823.65072
[6] Franklin, M.: Parallel solution of ordinary differential equations. IEEE Transactions on ComputersC-27, 413–420 (1978). · Zbl 0379.68033
[7] Gear, C.: Parallel methods for ordinary differential equations. Research R-87-1369, University of Illinois, Urbana, IL, 1986. · Zbl 0675.65068
[8] Hairer, E., Norsett, S., Wanner, G.: Solving ordinary differential equations. I. Nonstiff problems, vol. 1. Berlin, Heidelberg: Springer 1987. · Zbl 0638.65058
[9] Hairer, E., Wanner, G.: Solving ordinary differential equations. II. Stiff and differential-algebraic problems, vol. 2. Berlin, Heidelberg, New York, Tokyo: Springer 1991. · Zbl 0729.65051
[10] Hindmarsh, A.: LSODE and LSODI, two new initial value ordinary equation solvers. ACM/SIGNUM Newsletter15, 10–11 (1980).
[11] Lefever, R., Nicolis, G.: Chemical instabilities and sustained oscillations. J. Theor. Biol.30, 267–284 (1971). · Zbl 1170.92344
[12] Ortega, J., Rheinbolt, W.: Iterative solution of nonlinear equations in several variables. New York, San Francisco, London: Academic Press, 1970.
[13] 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). · Zbl 0309.65034
[14] Shampine, L., Watts, H.: A-stable implicit one-step methods. BIT12, 252–266 (1972). · Zbl 0253.65045
[15] Sommeijer, B., Couzy, W., Houwen, P. van der: A-stable parallel block methods for ordinary and integro-differential equations. PhD thesis, Universiteit van Amsterdam, CWI, Amsterdam, 1992. · Zbl 0749.65050
[16] Houwen, P. van der, Sommeijer, B.: Iterated Runge-Kutta methods on parallel computers. SIAM J. Sci. Statist. Comput.12, 1000–1028 (1991). · Zbl 0732.65065
[17] Vermiglio, R.: Parallel step methods for difference and differential equations. Tech. Rep., C.N.R. Progetto Finaizzato ”Sistemi Informatici e Calcolo Parallelo”, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.