On the stability and accuracy of onestep methods for solving stiff systems of ordinary differential equations
Authors:
A. Prothero and A. Robinson
Journal:
Math. Comp. 28 (1974), 145162
MSC:
Primary 65L05
MathSciNet review:
0331793
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Abstract: The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form The stability and accuracy of numerical approximations to the solution , obtained using implicit onestep integration methods, are studied. An Sstability property is introduced for this problem, generalizing the concept of Astability. A set of stiffly accurate onestep methods is identified and the concept of stiff order is defined in the limit . These additional properties are enumerated for several classes of Astable onestep methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initialvalue 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.
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 [1]
 G. G. Dahlquist, "A special stability problem for linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 3, 1963, pp. 2743. 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. 6570. MR 35 #6373. MR 0215533 (35:6373)
 [3]
 S. P. Norsett, "A criterion for stability of linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 259263. 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, NorthHolland, Amsterdam, 1969, pp. 187193. MR 41 #4808. MR 0260180 (41:4808)
 [5]
 C. E. Treanor, "A method for the numerical integration of coupled firstorder differential equations with greatly different time constants," Math. Comp., v. 20, 1966, pp. 3945. MR 33 #889. MR 0192664 (33:889)
 [6]
 S. P. Norsett, An AStable Modification of the AdamsBashforth Methods, Conf. on the Numerical Solution of Differential Equations (Dundee, Scotland, June 1969), Springer, Berlin, 1969, pp. 214219. MR 42 #2673. MR 0267771 (42:2673)
 [7]
 B. L. Ehle, "Highorder Astable methods for the numerical solution of systems of differential equations," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 276278. MR 39 #1119. MR 0239762 (39:1119)
 [8]
 B. L. Ehle, On Padé Approximations to the Exponential Function and AStable 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 Astable methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 185199. MR 40 #8266. MR 0255059 (40:8266)
 [10]
 O. Axelsson, "A note on a class of strongly Astable methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 14. MR 0315896 (47:4445)
 [11]
 F. H. Chipman, "Astable RungeKutta processes," Nordisk Tidskr. Informationsbehandling (BIT), v. 11, 1971, pp. 384388. MR 45 #4648. MR 0295582 (45:4648)
 [12]
 F. H. Chipman, Numerical Solution of Initial Value Problems Using AStable RungeKutta Processes, Report CSRR 2042, University of Waterloo, Department of Applied Analysis and Computer Science, June 1971.
 [13]
 H. A. Watts & L. F. Shampine, "Astable block implicit onestep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 252266. MR 0307483 (46:6603)
 [14]
 M. P. Halstead, A. Prothero & C. P. Quinn, "A mathematical model of the coolflame oxidation of acetaldehyde," Proc. Roy. Soc. London Ser. A, v. 322, 1971, pp. 377403.
 [15]
 J. C. Butcher, "Implicit RungeKutta processes," Math. Comp., v. 18, 1964, pp. 5064. 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. 266275.
 [17]
 A. R. Gourlay, "A note on trapezoidal methods for the solution of initial value problems," Math. Comp., v. 24, 1970, pp. 629633. MR 43 #1433. MR 0275680 (43:1433)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403317932
PII:
S 00255718(1974)03317932
Keywords:
Stiff system of ordinary differential equations,
implicit onestep methods,
Astability,
Sstability,
stiffly accurate methods,
stiff order
Article copyright:
© Copyright 1974
American Mathematical Society
