An improved class of generalized Runge–Kutta methods for stiff problems. Part II: The separated system case

摘要

In order to study a generalization of the explicit p-stage Runge–Kutta methods, we introduce a new family of p-stage formulas for the numerical integration of some special systems of ODEs that provides better order and stability results with the same number of stages. In our recent paper entitled `An improved class of generalized Runge–Kutta methods for stiff problems. Part I: The scalar case' we studied new schemes for the numerical integration of scalar autonomous ODEs. In this second part we will show that it is possible to generalize our GRK-methods so that they can be applied to some non-autonomous scalar ODEs and systems obtaining linearly implicit A-stable and L-stable methods. These methods do not require Jacobian evaluations in their implementation. Some numerical examples are discussed in order to show the good performance of the new schemes.