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

Abstract

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 d y / d x = − P y + Q ( x ) dy/dx = - Py + Q(x) . The method is exact if the differential equation is of the form shown, where P P is constant and Q ( x ) Q(x) is a quadratic in x x . The algorithm utilizes only the first derivative and thus has a parallel to the Runge-Kutta method. For P h Ph small (where h h is the integration interval) the method is identical to fourth-order Runge-Kutta and thus is correct to order h 4 {h^4} . Results for the coupled chemistry of high-temperature air are compared with results obtained from the usual Runge-Kutta procedure.