*(English)*Zbl 0954.65058

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. *J. C. Butcher* and *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.

Numerical experiments show the efficiency of the proposed methods.

##### MSC:

65L06 | Multistep, Runge-Kutta, and extrapolation methods |

65L05 | Initial value problems for ODE (numerical methods) |

34A34 | Nonlinear ODE and systems, general |

65L20 | Stability and convergence of numerical methods for ODE |