A new step-size changing technique for multistep methods
Authors:
G. K. Gupta and C. S. Wallace
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
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Abstract | References | Similar Articles | Additional Information
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.
- 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 https://doi.org/10.1145/355626.355636 W. H. ENRIGHT, T. E. HULL & B. LINDBERG, (1975), "Comparing numerical methods for stiff systems of O.D.E.s," BIT, v. 15, pp. 10-48.
- 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 https://doi.org/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 https://doi.org/10.1137/0711079 G. K. GUPTA, (1975), New Multistep Methods for the Solution of Ordinary Differential Equations, Ph. D. Thesis, Monash Univ., Australia. (Unpublished.)
- G. K. Gupta, Some new high-order multistep formulae for solving stiff equations, Math. Comp. 30 (1976), no. 135, 417–432. MR 423812, DOI https://doi.org/10.1090/S0025-5718-1976-0423812-1 G. K. GUPTA (1978), Numerical Testing of the ASI Technique of Step-Size Changing, Tech. Rep., Dept. of Computer Science, Monash Univ., Victoria, Australia.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1137/0713002
- Fred T. Krogh, Algorithms for changing the step size, SIAM J. Numer. Anal. 10 (1973), 949–965. MR 356515, DOI https://doi.org/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 https://doi.org/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), An Effective Variable Order Variable Step Adams Method, Tech. Rep. No. 53, Dept. of Computer Science, Univ. of Toronto, Canada.
- 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 https://doi.org/10.1016/0096-3003%2875%2990030-2
- 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), 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.
- 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
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Additional Information
Keywords:
Multistep methods,
stiff equations,
ordinary differential equations,
initial value problems
Article copyright:
© Copyright 1979
American Mathematical Society