On Numerical Integration of Ordinary Differential Equations

A reliable efficient general-purpose method for automatic digital com- puter integration of systems of ordinary differential equations is described. The method operates with the current values of the higher derivatives of a polynomial approximating the solution. It is thoroughly stable under all circumstances, in- corporates automatic starting and automatic choice and revision of elementary interval size, approximately minimizes the amount of computation for a specified accuracy of solution, and applies to any system of differential equations with deriva- tives continuous or piecewise continuous with finite jumps. ILLIAC library sub- routine # F7, University of Illinois Digital Computer Laboratory, is a digital computer program applying this method. 1. Introduction. A typical common scientific application of automatic digital computers is the integration of systems of ordinary differential equations. The author has developed a general-purpose method for doing this and explains the method here. While it is primarily designed to optimize the efficiency of large-scale calculations on automatic computers, its essential procedures also lend themselves well to hand computation. The method has the following characteristics, all of which are requisite to a satisfactory general-purpose method: a. Thorough stability with a large margin of safety under all circumstances. (Instabilities in the subject differential equations themselves are, of course, re- flected in the solution, but no further instabilities are introduced by the numerical procedures.) b. Any integration is started with only the essential initial conditions, i.e. there is a built-in automatic starting procedure. c. An optimum elementary interval size is automatically chosen, and the choice is automatically revised either upward or downward in the course of an integration, to provide the specified accuracy of solution in the minimum number of elementary steps. d. The derivatives need be computed just twice per elementary step, which is the minimum consistent with controlling accuracy. e. Any system of equations