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Time-point relaxation Runge-Kutta methods for ordinary differential equations. (English) Zbl 0783.65063

Time-point relaxation Runge-Kutta methods are implemented in Gauss-Jacobi and Gauss-Seidel modes. The authors show that if the number of Picard- Lindelöf iterations tends to infinity, then these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta method. The convergence order and the stability regions are investigated in detail.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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References:

[1] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Stability analysis of time-point relaxation Heun method, Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, 1, 30 (1990), Report
[2] A. Bellen, Z. Jackiewicz and M. Zennaro, Contractivity of waveform relaxation Runge-Kutta methods for dissipative systems in the maximum norm, SIAM J. Numer. Anal., to appear; A. Bellen, Z. Jackiewicz and M. Zennaro, Contractivity of waveform relaxation Runge-Kutta methods for dissipative systems in the maximum norm, SIAM J. Numer. Anal., to appear · Zbl 0818.65067
[3] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Time-point relaxation Runge-Kutta methods for ordinary differential equations, Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, 1, 75 (1991), Report
[4] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods (1987), Wiley: Wiley New York · Zbl 0616.65072
[5] Lelarasmee, E., The waveform relaxation method for the time domain analysis of large scale nonlinear dynamical system, (Ph.D. Thesis (1982), Univ. California: Univ. California Berkeley)
[6] Lelarasmee, E.; Ruehli, A. E.; Sangiovanni-Vincentelli, A., The waveform relaxation method for time domain analysis of large scale integrated circuits, IEEE Trans. Comput.-Aided Design Integrated Circuits System, 1, 131-145 (1982)
[7] Lie, I.; Skålin, R., Relaxation based integration by Runge-Kutta methods and its applications to the moving finite element method (1989), manuscript
[8] Nevanlinna, O., Remarks on Picard-Lindelöf iteration, BIT, 29, 328-346 (1988), Part I · Zbl 0673.65037
[9] Nevalinna, O., Remarks on Picard-Lindelöf iteration, BIT, 29, 535-562 (1989), Part II · Zbl 0697.65057
[10] White, J.; Sangiovanni-Vincentelli, A.; Odeh, F.; Ruehli, A., Waveform relaxation: Theory and practise, Trans. Soc. Comput. Simulation, 2, 95-133 (1985)
[11] Zennaro, M., Natural continuous extensions of Runge-Kutta methods, Math. Comp., 46, 305-318 (1986) · Zbl 0608.65043
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