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Type-insensitive ODE codes based on implicit $ A(\alpha )$-stable formulas


Author: L. F. Shampine
Journal: Math. Comp. 39 (1982), 109-123
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1982-0658216-2
MathSciNet review: 658216
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Abstract | References | Similar Articles | Additional Information

Abstract: Previous work on A-stable formulas is extended to $ A(\alpha )$-stable formulas, which are far more important in practice. Some important improvements in technique based on another interation method and an idea of Enright for the efficient handling of Jacobians are proposed. Implementation details and numerical examples are provided for a research-grade code.


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

  • [1] W. H. Enright, "Improving the efficiency of matrix operations in the numerical solution of stiff ordinary differential equations," ACM Trans. Math. Software, v. 4, 1978, pp. 127-136. MR 0483482 (58:3483)
  • [2] W. H. Enright, T. E. Hull & B. Lindberg, "Comparing numerical methods for stiff systems of O.D.E's," BIT, v. 15, 1975, pp. 10-48.
  • [3] A. C. Hindmarsh, "LSODE and LSODI, two new initial value ordinary differential equation solvers," SIGNUM Newsletter, v. 15, 1980, pp. 10-11.
  • [4] P. J. van der Houven, Construction of Integration Formulas for Initial Value Problems, North-Holland, Amsterdam, 1977. MR 0519726 (58:24960)
  • [5] L. F. Shampine, "Stiffness and non-stiff differential equation solvers," in Numerische Behandlung von Differentialgleichungen (R. Ansorge et al., eds.), Birkhauser, Basel, 1975. MR 0408261 (53:12026)
  • [6] L. F. Shampine, "Evaluation of implicit formulas for the solution of ODEs," BIT, v. 19, 1979, pp. 495-502. MR 559959 (81h:65079)
  • [7] L. F. Shampine, "Implementation of Rosenbrock methods," ACM Trans. Math. Software. (To appear.) MR 661123 (83f:65115)
  • [8] L. F. Shampine, "Implementation of implicit formulas for the solution of ODEs," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 103-118. MR 572543 (81e:65041)
  • [9] L. F. Shampine, "Evaluation of a test set for stiff ODE solvers," ACM Trans. Math. Software., v. 7, 1981, pp. 409-420. MR 654902 (83d:65215)
  • [10] L. F. Shampine, "Type-insensitive ODE codes based on implicit A-stable formulas," Math. Comp., v. 36, 1981, pp. 499-510. MR 658216 (83f:65116)
  • [11] L. F. Shampine & M. K. Gordon, Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman, San Francisco. Calif., 1975. MR 0478627 (57:18104)
  • [12] L. F. Shampine & H. A. Watts, DEPAC--Design of a User Oriented Package of ODE Solvers, Report SAND79-2374, Sandia National Laboratories, Albuquerque, N. M., 1980.
  • [13] S. Skelboe, "The control of order and steplength for backward differentiation formulas," BIT, v. 17, 1977, pp. 91-107. MR 0451740 (56:10022)
  • [14] A. Prothero, "Introduction to stiff problems," Chap. 9 in Modern Numerical Methods for Ordinary Differential Equations (G. Hall and J. M. Watt, eds.), Clarendon Press, Oxford, 1976.
  • [15] J. Williams, The Problem of Implicit Formulas in Numerical Methods for Stiff Differential Equations, Report 40, Dept. of Math., Univ. of Manchester, Manchester, England, 1979.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0658216-2
Keywords: ODE codes, stiffness, $ A(\alpha )$-stable
Article copyright: © Copyright 1982 American Mathematical Society

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