A new type of singly-implicit Runge-Kutta method

Implicit Runge-Kutta methods are considered which combine the single-implicitness or diagonal-implicitness property with a zero first row in the coefficient matrix. Acceptable stability for stiff problems is retained by requiring the last stage of a step to be identical to the output value. This requirement, which corresponds to the FSAL property for explicit Runge-Kutta methods, allows the method to have one less SIRK stage to achieve a specific stage order. Examples are given of DIRK, SIRK as well as DESI methods modified in this way. Methods are also proposed which have less than the full stage-order compared with the overall order of the method. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.