An implicit two-point numerical integration formula for linear and nonlinear stiff systems of ordinary differential equations

Author:
Simeon Ola. Fatunla

Journal:
Math. Comp. **32** (1978), 1-11

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1978-0474830-0

MathSciNet review:
0474830

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Abstract | References | Similar Articles | Additional Information

Abstract: In [1], the author proposed a semi-implicit one-step integration formula which effectively copes with linear systems of ordinary differential equations with widely varying eigenvalues. The integration algorithm is based on a local representation of the theoretical solution to the initial value problem by a linear combination of exponential functions. The resultant integration formula is of order four. Unfortunately, this algorithm cannot cope with nonlinear stiff systems of ordinary differential equations. In this paper, the author extends the concept adopted in [1] to construct an implicit two-point formula which can effectively cope with nonlinear stiff systems. The resultant integration formula is of order five and it is *L*-stable and convergent.

**[1]**S. O. FATUNLA, "A new semi-implicit integration algorithm to cope with stiff systems of ordinary differential equations," Manuscript, 1976.**[2]**K. M. BROWN, "A quadratically convergent Newton-like method based upon Gaussian elimination,"*Math. Comp.*, v. 20, 1966, pp. 11-20.**[3]**J. D. Lambert,*Nonlinear methods for stiff systems of ordinary differential equations*, Conference on the Numerical Solution of Differential Equations (Univ. of Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 75–88. Lecture Notes in Math., Vol. 363. MR**0426436****[4]**Leon Lapidus and John H. Seinfeld,*Numerical solution of ordinary differential equations*, Mathematics in Science and Engineering, Vol. 74, Academic Press, New York-London, 1971. MR**0281355****[5]**M. E. Fowler and R. M. Warten,*A numerical integration technique for ordinary differential equations with widely separated eigenvalues*, IBM J. Res. Develop**11**(1967), 537–543. MR**0216757**, https://doi.org/10.1147/rd.115.0537**[6]**Werner Liniger and Ralph A. Willoughby,*Efficient integration methods for stiff systems of ordinary differential equations*, SIAM J. Numer. Anal.**7**(1970), 47–66. MR**0260181**, https://doi.org/10.1137/0707002

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0474830-0

Keywords:
Stiff systems,
ordinary differential equations,
stiffness parameters,
*A*-stable,
*L*-stable,
interpolating functions,
eigenvalues,
stepsize,
Lipschitz constants

Article copyright:
© Copyright 1978
American Mathematical Society