Limiting precision in differential equation solvers. II. Sources of trouble and starting a code

Author:
L. F. Shampine

Journal:
Math. Comp. **32** (1978), 1115-1122

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1978-0501936-X

MathSciNet review:
0501936

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Abstract | References | Similar Articles | Additional Information

Abstract: The reasons a class of codes for solving ordinary differential equations might want to use an extremely small step size are investigated. For this class the likelihood of precision difficulties is evaluated and remedies examined. The investigation suggests a way of selecting automatically an initial step size which should be reliably on scale.

**[1]**L. F. SHAMPINE, "Limiting precision in differential equation solvers,"*Math. Comp.*, v. 28, 1974, pp. 141-144. MR**0329308 (48:7650)****[2]**L. F. SHAMPINE, H. A. WATTS & S. M. DAVENPORT, "Solving nonstiff ordinary differential equations--the state of the art,"*SIAM Rev.*, v. 18, 1976, pp. 376-411. MR**0413522 (54:1636)****[3]**L. F. SHAMPINE & H. A. WATTS, "Comparing error estimators for Runge-Kutta methods,"*Math. Comp.*, v. 25, 1971, pp. 445-455. MR**0297138 (45:6196)****[4]**L. F. SHAMPINE & M. K. GORDON,*Computer Solution of Ordinary Differential Equations*, Freeman, San Francisco, Calif., 1975. MR**0478627 (57:18104)****[5]**L. F. SHAMPINE,*Starting an ODE Solver*, Report SAND 77-1023, Sandia Laboratories, Albuquerque, N. M., 1977.**[6]**G. D. BYRNE & A. C. HINDMARSH, "A polyalgorithm for the numerical solution of ordinary differential equations,"*ACM Trans. Math. Software*, v. 1, 1975, pp. 71-98. MR**0378432 (51:14600)****[7]**G. J. LASTMAN, R. A. WENTZELL & A. C. HINDMARSH, "Numerical solution of a bubble cavitation problem,"*J. Computational Phys.*(To appear.) MR**0502908 (58:19804)****[8]**A. E. SEDGWICK,*An Effective Variable Order Variable Step Adams Method*, Report 53, Dept. of Comp. Sci., Univ. of Toronto, Toronto, Canada, 1973.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0501936-X

Keywords:
Limiting precision,
minimum step size,
roundoff,
initial step size

Article copyright:
© Copyright 1978
American Mathematical Society