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

Author:
Charles E. Treanor

Journal:
Math. Comp. **20** (1966), 39-45

MSC:
Primary 65.61

DOI:
https://doi.org/10.1090/S0025-5718-1966-0192664-3

MathSciNet review:
0192664

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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 . The method is exact if the differential equation is of the form shown, where is constant and is a quadratic in . The algorithm utilizes only the first derivative and thus has a parallel to the Runge-Kutta method. For small (where is the integration interval) the method is identical to fourth-order Runge-Kutta and thus is correct to order . Results for the coupled chemistry of high-temperature air are compared with results obtained from the usual Runge-Kutta procedure.

**[1]***Mathematical methods for digital computers*, John Wiley & Sons, Inc., New York-London, 1960. MR**0117906****[2]**Stephen H. Crandall,*Engineering analysis; a survey of numerical procedures*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. MR**0083185****[3]**G. Emanuel,*Problems Underlying the Numerical Integration of the Chemical and Vibrational Rate Equations in a Near Equilibrium Flow*, Report AEDC-TDR-63-82, Arnold Engineering Development Center, Tullahoma, Tenn., 1963.**[4]**C. F. Curtiss and J. O. Hirschfelder,*Integration of stiff equations*, Proc. Nat. Acad. Sci. U. S. A.**38**(1952), 235–243. MR**0047404****[5]**P. V. Marrone,*Inviscid, Nonequilibrium Flow Behind Bow and Normal Shock Waves*,*Part*I.*General Analysis and Numerical Examples*, Report QM-1626-A-12(I), Cornell Aeronautical Laboratory, Buffalo, N.Y., 1963.**[6]**P. V. Marrone & L. J. Garr,*Inviscid, Nonequilibrium Flow Behind Bow and Normal Shock Waves*,*Part*II.*The IBM*704*Computer Programs*, Report QM-1626-A-12(II), Cornell Aeronautical Laboratory, Buffalo, N.Y., 1963.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1966-0192664-3

Article copyright:
© Copyright 1966
American Mathematical Society