On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations. (English) Zbl 0836.65087

This paper is concerned with the so called waveform relaxation methods for the numerical solution of initial value problems (IVPs) for systems of ordinary differential equations (ODEs) by using parallel computers. The basic idea in the waveform relaxation methods consists in splitting the original system of ODEs into several subsystems together with an iterative process so that in each iteration these subsystems can be solved independently and the whole iterative process converges to the solution of the IVP under consideration.
Here, the author deals with waveform relaxation methods for nonlinear stiff IVPs, where the solution of subsystems is carried out with algebraically stable Runge-Kutta methods on (possibly) non-uniform grids. By introducing suitable assumptions on the stiff differential system, the author derives a convergence result of the numerical approximations on each iterate to the Runge-Kutta solution on the grid points. Furthermore some estimates of the errors with respect to the exact solution of the iterates are derived in suitable discrete and continuous norms. Finally, some comparisons with related results given by Ch. Lubich and A. Ostermann [BIT 27, 216-234 (1987; Zbl 0623.65125)] on uniform grids are presented.
Reviewer: M.Calvo (Zaragoza)


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65Y05 Parallel numerical computation
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E13 Multiple scale methods for ordinary differential equations


Zbl 0623.65125
Full Text: DOI


[1] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Contractivity of waveform relaxation Runge-Kutta iterations and related limit methods for dissipative systems in the maximum norm, SIAM J. Numer. Anal., 31, 499-523 (1994) · Zbl 0818.65067
[2] Bellen, A.; Zennaro, M., The use of Runge-Kutta formulae in waveform relaxation methods, Appl. Numer. Math., 11, 95-114 (1993) · Zbl 0786.65058
[3] Burrage, K., Parallel methods for initial value problems, Appl. Numer. Math., 11, 5-25 (1993) · Zbl 0781.65060
[4] Burrage, K.; Butcher, J. C., Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16, 46-57 (1979) · Zbl 0396.65043
[5] Burrage, K.; Pohl, B., Implementing an ODE code on distributed memory computers (1993), Department of Mathematics, University of Queensland, Preprint · Zbl 0810.65073
[6] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations (1987), Wiley: Wiley Chichester · Zbl 0616.65072
[7] Crouzeix, M., Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math., 32, 75-82 (1979) · Zbl 0431.65052
[8] Crouzeix, M.; Hundsdorfer, W. H.; Spijker, M. N., On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT, 23, 84-91 (1983) · Zbl 0506.65030
[9] Dekker, K.; Verwer, J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0571.65057
[10] Gear, C. W., Massive parallelism across space in ODEs, Appl. Numer. Math., 11, 27-43 (1993) · Zbl 0798.65075
[11] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0729.65051
[12] Hundsdorfer, W. H., The numerical solution of nonlinear stiff initial value problems: an analysis of one step methods, (CWI Tract, 12 (1985), Centre for Mathematics and Computer Science: Centre for Mathematics and Computer Science Amsterdam) · Zbl 0557.65048
[14] Lelarasmee, E.; Ruehli, A. E.; Sangiovanni-Vincentelli, A. L., The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. CAD IC Syst., 1, 131-145 (1982)
[15] Lindelöf, E., Sur l’application des méthodes d’approximation successives à l’étude des intégrales réelles des équations différentielles ordinaires, J. Math. Pures Appl., 10, 117-128 (1894), (4e série) · JFM 25.0509.03
[16] Lùbich, C.; Ostermann, A., Multi-grid dynamic iteration for parabolic equations, BIT, 27, 216-234 (1987) · Zbl 0623.65125
[17] Miekkala, U.; Nevanlinna, O., Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Statist. Comput., 8, 459-482 (1987) · Zbl 0625.65063
[18] Miekkala, U.; Nevanlinna, O., Sets of convergence and stability regions, BIT, 27, 554-584 (1987) · Zbl 0633.65064
[19] Nevanlinna, O., Remarks on Picard-Lindelöf iteration, Part I, BIT, 29, 328-346 (1989) · Zbl 0673.65037
[20] Nevanlinna, O., Remarks on Picard-Lindelöf iteration, Part II, BIT, 29, 535-562 (1989) · Zbl 0697.65057
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