A. Classical Fourth Order Runge-Kutta
This method is an implementation of the fourth order Runge-Kutta method. A relative error term is used to adjust the step-size. Error control is achieved by simply halving the step size until the change in calculated values is smaller than the error requested [Kutta, 1901].
This is the numerical integration method used in MULTI(RUNGE) by Yamaoka et.al. It has also been modified to give automatic step-size control. Again, error control is achieved by halving the step size until the desired accuracy is achieved. In MULTI(RUNGE) the step size was adjusted manually [Gill, 1951].
This is a further modification of the Runge-Kutta fourth order method which uses a fifth order calculation to determine the appropriate step size given values for the required relative and absolute error. This is the subroutine written by Watts and Shampine and published as Subroutine RKF45 in the book by Forsythe, Malcolm, and Moler [Fehlberg, 1969; Watt, 1977; Shampine, 1977]. This method is very efficient and appears to be the method of choice for non-stiff systems of differential equations.
This is one of the options of the subroutine written by Gear. This method is also for non-stiff systems of differential equations. Step-size and order are automatically controlled to achieve a user defined absolute error [Gear, 1969; Gear, 1971a; Gear, 1971b].
E. Gear with PEDERV subroutine
This is the most efficient option for use with stiff equations. The subroutines DECOMP and SOLVE were modified from those presented in the text by Forsythe and Moler [Forsythe, 1967].
F. Gear without PEDERV subroutine
The partial derivatives are calculated by numerical differencing thus a user supplied PEDERV subroutine is not required.
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