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**Remarks on waveform relaxation method with overlapping splittings.**
*(English)*
Zbl 0907.65063

This paper is concerned with the strategy of overlapping splittings in using waveform relaxation methods on finite time intervals. Starting from the iteration operator and its resolvent, the order of the iteration operator turns out to be the decisive quantity for the valuation of the method. It is shown that this order may decrease if overlapping is used, and further, how overlapping should be done to best accelerate of the iterative method. The results are illustrated in treating band matrices.

Reviewer: R.Scherer (Karlsruhe)

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65F10 | Iterative numerical methods for linear systems |

34A30 | Linear ordinary differential equations and systems |

### Keywords:

digraph; quasinilpotency; overlapping splittings; waveform relaxation methods; iterative method; band matrices
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\textit{U. Miekkala}, J. Comput. Appl. Math. 88, No. 2, 349--361 (1998; Zbl 0907.65063)

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### References:

[1] | Chen, W. K., Applied Graph Theory, Graphs and Electrical Networks (1976), North-Holland: North-Holland Amsterdam |

[2] | Jeltsch, R.; Pohl, B., Waveform relaxation with overlapping splittings, SIAM J. Sci. Comput., 16, 40-49 (1995) · Zbl 0821.65049 |

[3] | Juang, F., Waveform methods for ordinary differential equations, (Ph.D. Thesis (January 1990), University of Illinois at Urbana-Champaign, Dept of Computer Science), Report No. UIUCDCS-R-90-1563 |

[4] | Lelarasmee, E.; Ruehli, A.; Sangiovanni-Vincentelli, A., The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. CAD, 1, 131-145 (1982) |

[5] | 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 |

[6] | Miekkala, U.; Nevanlinna, O., Quasinilpotency of the operators in Picard-Lindelöf iteration, Numer. Funct. Anal. Optim., 13, 203-221 (1992) · Zbl 0764.65040 |

[7] | Vainberg, M. M.; Trenogin, V. A., Theory of Branching of Solutions of Non-linear Equations (1974), Noordhoff: Noordhoff Leyden · Zbl 0274.47033 |

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