Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Abstract

Abstract. Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

Stability of generalized Runge–Kutta methods for stiff kinetics coupled differential equations

“…Moreover, Rosenbrock-type methods in which the exact Jacobian is no longer needed have been considered. The generalized Runge-Kutta methods [9,12,15] fall into this class. For an excellent survey of some of these methods the reader may be referred to [16].…”