Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Polynomial formulation of second derivative multistep methods

Authors: S. Kovvali and G. K. Gupta
Journal: Math. Comp. 38 (1982), 447-458
MSC: Primary 65L05
MathSciNet review: 645662
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Following the work of Enright [3] there has been interest in studying second derivative methods for solving stiff ordinary differential equations. Successful implementations of second derivative methods have been reported by Enright [3], Sacks-Davis [9], [10] and Addison[l].

Wallace and Gupta [13] have suggested a polynomial formulation of the usual first-derivative multistep methods. Recently Skeel [11] has shown the equivalence of several formulations of multistep methods. The work of Wallace and Gupta [13] was extended to second derivative methods by Gupta [8]. The present work includes results obtained regarding the stability and truncation error of second derivative methods using the polynomial formulation.

References [Enhancements On Off] (What's this?)

  • [1] C. A. Addison, Implementing a Stiff Method Based Upon the Second Derivative Formulas, Technical Report No. 130/79, Dept. of Computer Science, University of Toronto, Canada, 1979.
  • [2] R. L. Brown, "Some characteristics of implicit multistep multi-derivative integration formulas," SIAM J. Numer. Anal., v. 14, 1977, pp. 982-993. MR 0471264 (57:11001)
  • [3] W. H. Enright, "Second derivative multistep methods for stiff ordinary differential equations," SIAM J. Numer. Anal., v. 11, 1974, pp. 321-331. MR 0351083 (50:3574)
  • [4] W. H. Enright, "Optimal second derivative methods for stiff systems," in Stiff Differential Systems (R. A. Willoughby, Ed.), Plenum Press, New York, 1974, pp. 95-109. MR 0343619 (49:8359)
  • [5] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
  • [6] W. Liniger & R. A Willoughby, "Efficient integration methods for stiff systems of ordinary differential equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 47-66. MR 0260181 (41:4809)
  • [7] G. K. Gupta, New Multistep Methods for the Solution of Ordinary Differential Equations, Ph. D. Thesis, Dept. of Computer Science, Monash University, Victoria, Australia, 1975.
  • [8] G. K. Gupta, "Implementing second derivative multistep methods using the Nordsieck polynomial representation," Math. Comp., v. 32, 1978, pp. 13-18. MR 0478630 (57:18107)
  • [9] R. Sacks-Davis, "Solution of stiff ordinary differential equations by a second derivative method," SIAM J. Numer. Anal., v. 14, 1977, pp. 1088-1100. MR 0471323 (57:11059)
  • [10] R. Sacks-Davis, "Fixed leading coefficient implementation of SD-Formulas for stiff ODE's," ACM Trans. Math. Software, v. 6, 1980, pp. 540-562. MR 599976 (81k:65084)
  • [11] R. D. Skeel, "Equivalent forms of multistep formulas," Math. Comp., v. 33, 1979, pp. 1229-1250. MR 537967 (80j:65027)
  • [12] R. D. Skeel & A. K. Kong, "Blended linear multistep methods," ACM Trans. Math. Software, v. 3, 1977, pp. 326-345 (also Report UIUCDCS-R-76-800, Dept. of Computer Science, Univ. of Illinois). MR 0461922 (57:1904)
  • [13] C. S. Wallace & G. K. Gupta, "General linear multistep methods to solve ordinary differential equations," Austral. Comput. J., v. 5, 1973, pp. 62-69. MR 0362919 (50:15357)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L05

Retrieve articles in all journals with MSC: 65L05

Additional Information

Keywords: Second derivative methods, multistep methods, numerical solution of ordinary differential equations, stiff equations
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society