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Zbl 0954.65058
Butcher, J.C.; Chen, D.J.L.
A new type of singly-implicit Runge-Kutta method. (English)
[J] Appl. Numer. Math. 34, No.2-3, 179-188 (2000). ISSN 0168-9274

The authors discuss new type of implicit Runge-Kutta methods which combine the singly-implicitness or diagonal-implicitness property with a zero first row in the coefficient matrix of the method. The new feature considered in this paper is the use of a first stage identical to the input approximation (this means that the first row of the coefficient matrix is zero) and the last stage , identical to the output value (this means that the last row of the coefficient matrix is identical with the vector of output value coefficients). The derived methods preserve the feature of the FSAL-methods and also of the DESI-methods [cf. {\it J. C. Butcher} and {\it J. R. Cash}, SIAM J. Numer. Anal. 27, No.~3, 753-761 (1990; Zbl 0702.65072)]. It is shown that the derivative of the first internal stage for a singly-implicit Runge-Kutta method can be obtained from the previous step without any loss of performance. \par Numerical experiments show the efficiency of the proposed methods.
[Iulian Coroian (Baia Mare)]

MSC 2000:: *65L06 Multistep, Runge-Kutta, and extrapolation methods
65L05 Initial value problems for ODE (numerical methods)
34A34 Nonlinear ODE and systems, general
65L20 Stability of numerical methods for ODE

Keywords: singly-implicit Runge-Kutta methods; FSAL methods; DESI methods; initial value problems; stability of methods; numerical experiments

Citations: Zbl 0702.65072

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