A process for the step-by-step integration of differential equations in an automatic digital computing machine

Abstract

It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.

Economical Evaluation of Runge-Kutta Formulae

“…However, it will now be shown, following Gill [1], that it is slightly better to let Pn -(-L¡-) P,2, \a -n + r) \ a -n + r/ \a -n + r)…”

“…Gill [1] and Blum [2] have produced special versions of the Runge-Kutta fourth order method for the solution of N simultaneous first order differential equations which require 3A 4-P storage locations against the normal 4JV -f P, where P is the storage required by the program. It is shown below that it is possible to arrange all such methods in a form which requires SN -\-P storage locations.…”

Bessel-function integrals needed for two classes of physics problems

“…Although we have not yet found an analytical proof, the numerical result very strongly suggests that in which the generating integrals were evaluated by Gill's method [7]. Various step sizes were tried and the results were checked against standard tables [8].…”

A modification of the Runge-Kutta fourth-order method

“…We now let y be the vector, (ya, yi, ■ ■ • , yn), and / the vector-valued function, (/0, /i , • • • , /"). The initial value problem can then be written as (1)(2)(3) y'=f(y), (1.4) y(k) = a.…”

“…A variant of this method was derived by S. Gill [1]. The two advantages of Gill's variant are (1) in automatic computers, it requires 'in -4-B storage registers whereas the Runge-Kutta formulas as given above, require An -f-B, where B is some constant; (2) the computation can be arranged so that rounding errors are reduced appreciably.…”