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On Turan type implicit Runge-Kutta methods

Runge-Kutta-Methoden über Turanschen Quadraturformeln

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Summary

Turan[5] has shown, that for a quadrature formula with multiple nodes

$$\int\limits_{x_0 }^{x_0 + h} {f(t)dt\dot = h\sum {c_i^{(1)} } f(x_0 + b_i h) + h^2 \sum {c_i^{(2)} f'(x_0 + b_i h) + ... + h^m \sum {c_i^{(m)} } ^{f(m - 1)} (x_0 + b_i h)} } $$

there exist, form odd, “Gaussian” nodesb 1, ...,b s, so that the quadrature formula reaches order (m+1)s. In the present paper we show that these formulas can be extended to Implicit Runge-Kutta methods with multiple nodes (cf. [4]) also of order (m+1)s, in the same way, asButcher's processes [1] generalize Gaussian formulas (casem=1).

Zusammenfassung

Turan hat in [5] gezeigt, daß bei einer Quadraturformel mit mehrfachen Knoten

$$\int\limits_{x_0 }^{x_0 + h} {f(t)dt\dot = h\sum {c_i^{(1)} } f(x_0 + b_i h) + h^2 \sum {c_i^{(2)} f'(x_0 + b_i h) + ... + h^m \sum {c_i^{(m)} } ^{f(m - 1)} (x_0 + b_i h)} } $$

beiungeradem m die Stützstellenb 1, ...,b s so gewählt werden können, daß die Methode die Ordnung (m+1)s erreicht. Wir zeigen hier, daß diese Formeln auf implizite RK-Methoden mit mehrfachen Knoten erweitert werden können, welche ebenfalls die Ordnung (m+1)s besitzen. Im Fallem=1 sind dies die Methoden vonButcher [1] über Gaußschen Quadraturformeln.

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References

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Authors and Affiliations

  1. Institut für Mathematik, Universität Innsbruck, A-6020, Innsbruck, Österreich

    K. Kastlunger &  G. Wanner

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  1. K. Kastlunger
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  2. G. Wanner
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Kastlunger, K., Wanner, G. On Turan type implicit Runge-Kutta methods. Computing 9, 317–325 (1972). https://doi.org/10.1007/BF02241605

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  • DOI: https://doi.org/10.1007/BF02241605

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