A predictorcorrector method for a certain class of stiff differential equations
Authors:
Karl G. Guderley and Chenchi Hsu
Journal:
Math. Comp. 26 (1972), 5169
MSC:
Primary 65M99
DOI:
https://doi.org/10.1090/S00255718197202989527
MathSciNet review:
0298952
Fulltext PDF Free Access
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 predictorcorrector 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 predictorcorrector 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.

K. G. Guderley & C. C. Hsu, “A special form of Galerkin’s method applied to heat transfer in plane CouettePoiseuille flows.” (In prep.)
 C. W. Gear, The automatic integration of stiff ordinary differential equations., Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) NorthHolland, Amsterdam, 1969, pp. 187–193. MR 0260180 D. A. Calahan, “Numerical solution of linear systems with widely separated time constants,” Proc. IEEE, v. 55, 1967, pp. 20162017.
 James L. Blue and Hermann K. Gummel, Rational approximations to matrix exponential for systems of stiff differential equations, J. Comput. Phys. 5 (1970), 70–83. MR 255060, DOI https://doi.org/10.1016/00219991%2870%29900537
 Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New YorkLondon, 1962. MR 0135729
 Fritz John, Lectures on advanced numerical analysis, Gordon and Breach Science Publishers, New YorkLondonParis, 1967. MR 0221721
 Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
 Lothar Collatz, Funktionalanalysis und numerische Mathematik, Die Grundlehren der mathematischen Wissenschaften, Band 120, SpringerVerlag, Berlin, 1964 (German). MR 0165651
 Richard S. Varga, Matrix iterative analysis, PrenticeHall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
 J. Certaine, The solution of ordinary differential equations with large time constants, Mathematical methods for digital computers, Wiley, New York, 1960, pp. 128–132. MR 0117917
Retrieve articles in Mathematics of Computation with MSC: 65M99
Retrieve articles in all journals with MSC: 65M99
Additional Information
Keywords:
Predictorcorrector method,
stiff differential equations,
interval control,
stability,
weighted infinity vector norm
Article copyright:
© Copyright 1972
American Mathematical Society