## A new step-size changing technique for multistep methods

HTML articles powered by AMS MathViewer

- by G. K. Gupta and C. S. Wallace PDF
- Math. Comp.
**33**(1979), 125-138 Request permission

## Abstract:

The step-size changing technique is an important component of a Variable Step Variable Order algorithm for solving ordinary differential equations using multi-step methods. This paper presents a new technique for changing the step-size and compares its performance to that of the Variable-Step and Fixed-Step Interpolation techniques.## References

- Robert K. Brayton, Fred G. Gustavson, and Gary D. Hachtel,
*A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas*, Proc. IEEE**60**(1972), 98–108. MR**0351101** - 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**378432**, DOI 10.1145/355626.355636
W. H. ENRIGHT, T. E. HULL & B. LINDBERG, (1975), "Comparing numerical methods for stiff systems of O.D.E.s," - C. William Gear,
*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898** - C. W. Gear,
*The automatic integration of ordinary differential equations*, Comm. ACM**14**(1971), no. 3, 176–179. MR**0388778**, DOI 10.1145/362566.362571 - C. W. Gear and K. W. Tu,
*The effect of variable mesh size on the stability of multistep methods*, SIAM J. Numer. Anal.**11**(1974), 1025–1043. MR**368436**, DOI 10.1137/0711079
G. K. GUPTA, (1975), - G. K. Gupta,
*Some new high-order multistep formulae for solving stiff equations*, Math. Comp.**30**(1976), no. 135, 417–432. MR**423812**, DOI 10.1090/S0025-5718-1976-0423812-1
G. K. GUPTA (1978), - G. K. Gupta and C. S. Wallace,
*Some new multistep methods for solving ordinary differential equations*, Math. Comp.**29**(1975), 489–500. MR**373290**, DOI 10.1090/S0025-5718-1975-0373290-5 - Peter Henrici,
*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729** - T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick,
*Comparing numerical methods for ordinary differential equations*, SIAM J. Numer. Anal.**9**(1972), 603–637; errata, ibid. 11 (1974), 681. MR**351086**, DOI 10.1137/0709052 - Rolf Jeltsch,
*Stiff stability and its relation to $A_{0}$- and $A(0)$-stability*, SIAM J. Numer. Anal.**13**(1976), no. 1, 8–17. MR**411174**, DOI 10.1137/0713002 - Fred T. Krogh,
*Algorithms for changing the step size*, SIAM J. Numer. Anal.**10**(1973), 949–965. MR**356515**, DOI 10.1137/0710081 - Fred T. Krogh,
*A variable step, variable order multistep method for the numerical solution of ordinary differential equations*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 194–199. MR**0261790** - Arnold Nordsieck,
*On numerical integration of ordinary differential equations*, Math. Comp.**16**(1962), 22–49. MR**136519**, DOI 10.1090/S0025-5718-1962-0136519-5 - Peter Piotrowski,
*Stability, consistency and convergence of variable $k$-step methods for numerical integration of large systems of ordinary differential equations*, Conf. on Numerical Solution of Differential Equations (Dundee, 1969) Springer, Berlin, 1969, pp. 221–227. MR**0277116**
A. SEDGWICK, (1973), - L. F. Shampine and M. K. Gordon,
*Local error and variable order Adams codes*, Appl. Math. Comput.**1**(1975), no. 1, 47–66. MR**373294**, DOI 10.1016/0096-3003(75)90030-2 - 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**
K. W. TU, (1972), - C. S. Wallace and G. K. Gupta,
*General linear multistep methods to solve ordinary differential equations*, Austral. Comput. J.**5**(1973), 62–69. MR**362919**

*BIT*, v. 15, pp. 10-48.

*New Multistep Methods for the Solution of Ordinary Differential Equations*, Ph. D. Thesis, Monash Univ., Australia. (Unpublished.)

*Numerical Testing of the ASI Technique of Step-Size Changing*, Tech. Rep., Dept. of Computer Science, Monash Univ., Victoria, Australia.

*An Effective Variable Order Variable Step Adams Method*, Tech. Rep. No. 53, Dept. of Computer Science, Univ. of Toronto, Canada.

*Stability and Convergence of General Multistep and Multivalue Methods with Variable Step Size*, Report No. UIUCDCS-R-72-526, Dept. of Computer Science, Univ. of Illinois, Unbana.

## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 125-138 - MSC: Primary 65L05; Secondary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1979-0514814-8
- MathSciNet review: 514814