On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations

Prothero, A.; Robinson, A.

doi:10.1090/S0025-5718-1974-0331793-2

On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations

Authors: A. Prothero and A. Robinson
Journal: Math. Comp. 28 (1974), 145-162
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1974-0331793-2
MathSciNet review: 0331793
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form

$\displaystyle y' = g'(x) + \lambda\{y - g(x)\}.$

The stability and accuracy of numerical approximations to the solution

, obtained using implicit one-step integration methods, are studied. An S-stability property is introduced for this problem, generalizing the concept of A-stability. A set of stiffly accurate one-step methods is identified and the concept of stiff order is defined in the limit $\operatorname{Re}(-\lambda) \to \infty$ . These additional properties are enumerated for several classes of A-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems.

[1] G. G. Dahlquist, "A special stability problem for linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 3, 1963, pp. 27-43. MR 30 #715. MR 0170477 (30:715)
[2] O. B. Widlund, "A note on unconditionally stable linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 7, 1967, pp. 65-70. MR 35 #6373. MR 0215533 (35:6373)
[3] S. P. Norsett, "A criterion for $A(\alpha )$ -stability of linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 259-263. MR 41 #1227. MR 0256571 (41:1227)
[4] C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations (With Discussion), Proc. IFIP Congress Information Processing 68 (Edinburgh 1968), vol. 1; Mathematics, Software, North-Holland, Amsterdam, 1969, pp. 187-193. MR 41 #4808. MR 0260180 (41:4808)
[5] C. E. Treanor, "A method for the numerical integration of coupled first-order differential equations with greatly different time constants," Math. Comp., v. 20, 1966, pp. 39-45. MR 33 #889. MR 0192664 (33:889)
[6] S. P. Norsett, An A-Stable Modification of the Adams-Bashforth Methods, Conf. on the Numerical Solution of Differential Equations (Dundee, Scotland, June 1969), Springer, Berlin, 1969, pp. 214-219. MR 42 #2673. MR 0267771 (42:2673)
[7] B. L. Ehle, "High-order A-stable methods for the numerical solution of systems of differential equations," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 276-278. MR 39 #1119. MR 0239762 (39:1119)
[8] B. L. Ehle, On Padé Approximations to the Exponential Function and A-Stable Methods for the Numerical Solution of Initial Value Problems, Report CSRR 2010, University of Waterloo, Department of Applied Analysis and Computer Science, March 1969.
[9] O. Axelsson, "A class of A-stable methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 185-199. MR 40 #8266. MR 0255059 (40:8266)
[10] O. Axelsson, "A note on a class of strongly A-stable methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 1-4. MR 0315896 (47:4445)
[11] F. H. Chipman, "A-stable Runge-Kutta processes," Nordisk Tidskr. Informationsbehandling (BIT), v. 11, 1971, pp. 384-388. MR 45 #4648. MR 0295582 (45:4648)
[12] F. H. Chipman, Numerical Solution of Initial Value Problems Using A-Stable Runge-Kutta Processes, Report CSRR 2042, University of Waterloo, Department of Applied Analysis and Computer Science, June 1971.
[13] H. A. Watts & L. F. Shampine, "A-stable block implicit one-step methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 252-266. MR 0307483 (46:6603)
[14] M. P. Halstead, A. Prothero & C. P. Quinn, "A mathematical model of the cool-flame oxidation of acetaldehyde," Proc. Roy. Soc. London Ser. A, v. 322, 1971, pp. 377-403.
[15] J. C. Butcher, "Implicit Runge-Kutta processes," Math. Comp., v. 18, 1964, pp. 50-64. MR 28 #2641. MR 0159424 (28:2641)
[16] J. H. Seinfeld, L. Lapidus & M. Hwang, "Review of numerical integration techniques for stiff ordinary differential equations," Ind. Eng. Chem. Fundamentals, v. 9, 1970, pp. 266-275.
[17] A. R. Gourlay, "A note on trapezoidal methods for the solution of initial value problems," Math. Comp., v. 24, 1970, pp. 629-633. MR 43 #1433. MR 0275680 (43:1433)