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

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.**28**(1974), 141–144. MR**0329308**, 10.1090/S0025-5718-1974-0329308-8**[2]**L. F. Shampine, H. A. Watts, and S. M. Davenport,*Solving nonstiff ordinary differential equations—the state of the art*, SIAM Rev.**18**(1976), no. 3, 376–411. MR**0413522****[3]**L. F. Shampine and H. A. Watts,*Comparing error estimators for Runge-Kutta methods*, Math. Comp.**25**(1971), 445–455. MR**0297138**, 10.1090/S0025-5718-1971-0297138-9**[4]**L. F. Shampine and M. K. Gordon,*Computer solution of ordinary differential equations*, W. H. Freeman and Co., San Francisco, Calif., 1975. The initial value problem. MR**0478627****[5]**L. F. SHAMPINE,*Starting an ODE Solver*, Report SAND 77-1023, Sandia Laboratories, Albuquerque, N. M., 1977.**[6]**G. D. Byrne and A. C. Hindmarsh,*A polyalgorithm for the numerical solution of ordinary differential equations*, ACM Trans. Math. Software**1**(1975), no. 1, 71–96. MR**0378432****[7]**G. J. Lastman, R. A. Wentzell, and A. C. Hindmarsh,*Numerical solution of a bubble cavitation problem*, J. Comput. Phys.**28**(1978), no. 1, 56–64. MR**0502908****[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