A METHOD FOR CONSTRUCTING GENERALIZED RUNGE-KUTTA METHODS

In the implementation of an implicit Runge-Kutta formula, we need to solve systems of nonlinear equations. In this paper, we analyze the Newton iteration process and a modified Newton iteration process for solving these equations. Then we propose the methods in which we take only a fixed finite number of iterations and adopt the last iterate as the approximate solution for the nonlinear equations. Our methods are a kind of generalized Runge-Kutta methods with the same order as the original Runge-Kutta formula and inherit its linear stability properties of the original implicit Runge-Kutta formula, for example, AN-stability, L-stability and S-stability. Based on this fact, we construct three methods of order five imbedding a fourth-order formula for error estimation. Finally, test results for 25 stiff problems are discussed.