A predictor-corrector method for a certain class of stiff differential equations

Authors:
Karl G. Guderley and Chen-chi Hsu

Journal:
Math. Comp. **26** (1972), 51-69

MSC:
Primary 65M99

DOI:
https://doi.org/10.1090/S0025-5718-1972-0298952-7

MathSciNet review:
0298952

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In stiff systems of linear ordinary differential equations, certain elements of the matrix describing the system are very large. Sometimes, e.g., in treating partial differential equations, the problem can be formulated in such a manner that large elements appear only in the main diagonal. Then the elements causing stiffness can be taken into account analytically. This is the basis of the predictor-corrector method presented here. The truncation error can be estimated in terms of the difference between predicted and corrected values in nearly the same manner as for the customary predictor-corrector method. The question of stability, which is crucial for stiff equations, is first studied for a single equation; as expected, the method is much more stable than the usual predictor- corrector method. For systems of equations, sufficient conditions for stability are derived which require less work than a detailed stability analysis. The main tool is a matrix norm which is consistent with a weighted infinity vector norm. The choice of the weights is critical. Their determination leads to the question whether a certain matrix has a positive inverse.

**[1]**K. G. Guderley & C. C. Hsu, ``A special form of Galerkin's method applied to heat transfer in plane Couette-Poiseuille flows.'' (In prep.)**[2]**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)****[3]**D. A. Calahan, ``Numerical solution of linear systems with widely separated time constants,''*Proc. IEEE*, v. 55, 1967, pp. 2016-2017.**[4]**J. L. Blue & H. K. Gummel, ``Rational approximations to matrix exponential for systems of stiff differential equations,''*J. Computational Physics*, v. 5, 1970, pp. 70-83. MR**40**#8267. MR**0255060 (40:8267)****[5]**P. Henrici,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley, New York, 1962. MR**24**#B1772. MR**0135729 (24:B1772)****[6]**F. John,*Lectures on Advanced Numerical Analysis*, Gordon and Breach, New York, 1967. MR**36**#4773. MR**0221721 (36:4773)****[7]**M. Marden,*Geometry of Polynominals*, 2nd ed., Math. Surveys, no. 3, Amer. Math. Soc., Providence, R. I., 1966. MR**37**#1562. MR**0225972 (37:1562)****[8]**L. Collatz,*Funktionalanalysis und numerische Mathematik*, Die Grundlehren der math. Wissenschaften, Band 120, Springer-Verlag, Berlin, 1964; English transl., Academic Press, New York, 1966. MR**29**#2931; MR**34**#4961. MR**0165651 (29:2931)****[9]**R. S. Varga,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**28**#1725. MR**0158502 (28:1725)****[10]**J. Certaine, ``The solution of ordinary differential equations with large time constants,'' in*Mathematical Methods for Digital Computers*, A. Ralston and H. S. Wilf (Editors), Wiley, New York, 1960, pp. 128-132. MR**22**#8691. MR**0117917 (22:8691)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M99

Retrieve articles in all journals with MSC: 65M99

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0298952-7

Keywords:
Predictor-corrector method,
stiff differential equations,
interval control,
stability,
weighted infinity vector norm

Article copyright:
© Copyright 1972
American Mathematical Society