# On Turan type implicit Runge-Kutta methods

@article{Kastlunger1972OnTT,
title={On Turan type implicit Runge-Kutta methods},
author={K. H. Kastlunger and Gerhard Wanner},
journal={Computing},
year={1972},
volume={9},
pages={317-325}
}
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… CONTINUE READING

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