A new type of singly-implicit Runge–Kutta method Original Research Article

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.