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Coupled differential equations which describe the simultaneous relaxation of different components at greatly different rates present a difficulty in numerical integration, since the integration interval is determined by the fastest rate, and the region of integration is determined by the slowest rate. In the present paper an integration formula is derived from the approximation that within an interval the first derivative can be expressed as dy/dx = -Py + Q(x). The method is exact if the differential equation is of the form shown, where P is constant and Q(x) is a quadratic in x. The algorithm utilizes only the first derivative and thus has a parallel to the Runge-Kutta method. For Ph small (where h is the integration interval) the method is identical to fourth-order Runge-Kutta and thus is correct to order h4. Results for the coupled chemistry of high-temperature air are compared with results obtained from the usual Runge-Kutta procedure. 1. Introduction. Of the many methods available for the numerical solution of ordinary differential equations or sets of coupled ordinary differential equations (1), (2) most have specific advantages so that they are chosen for special types of problems. Several methods have a general applicability that has made them popular in a wide variety of cases. There exists, however, a class of equations which present considerable difficulty in their solution and for which there does not seem to be a simple and accurate method available. These equations can be characterized by the fact that the derivative of a dependent variable has a strong dependence on the difference between its own value and that of a slowly varying function. Because of this strong dependence they are sometimes referred to as "stiff" equations (3), (4). An example is the simultaneous relaxation of several coupled components with widely different relaxation times, where the calculation interval for the entire relaxation region is determined by the time constant of the most rapid process. In problems of high-temperature air flows, the chemical rate equations some- times have this character, involving both fast and slow chemical rates in some regions of integration. Similar problems appear in the calculation of both vibrational relaxation and electronic excitation during the dissociation of diatomic molecules. It is desirable to have a method of integration which can handle the stiff-equation problem when it arises, but which will provide the proper speed and accuracy for those portions of the calculation where no special treatment is required. In the present report a method of integration is derived for the specific purpose of handling this problem. In a case where the derivative is strongly dependent on the value of the dependent variable, the integration formula takes this dependence into account directly and does away with the strong oscillations generated by usual integration formulas such as Runge-Kutta's. However, in the case where the derivative does not have this strong dependence, the method becomes identical

A method for the numerical integration of coupled first-order differential equations with greatly different time constants