Two Classes of Explicit Generalized Runge-Kutta Processes for Non-Stiff Systems of Ordinary Differential Equations

The computation of non-stiff systems of ordinary differential equations can be accomplished with explicit Runge-Kutta methods. A class of explicit Generalized Runge-Kutta is described which requires an accurate evaluation of a Jacobian at every step. Second and fourth order processes are also described. In addition a second class of explicit Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated less than once every step. Finally a third order process is described. It is shown that these methods lend themselves easily to the development of error estimators similar to those of Fehlberg or England