Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



A necessary condition for $ A$-stability of multistep multiderivative methods

Author: Rolf Jeltsch
Journal: Math. Comp. 30 (1976), 739-746
MSC: Primary 65L05
MathSciNet review: 0431690
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The region of absolute stability of multistep multiderivative methods is studied in a neighborhood of the origin. This leads to a necessary condition for A-stability. For methods where $ \rho (\zeta )/(\zeta - 1)$ has no roots of modulus 1 this condition can be checked very easily. For Hermite interpolatory and Adams type methods a necessary condition for A-stability is found which depends only on the error order and the number of derivatives used at $ ({x_{n + k}},{y_{n + k}})$.

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

  • [1] R. L. BROWN, "Multi-derivative numerical methods for the solution of stiff ordinary differential equations," Dept. of Comput. Sci., Univ. of Illinois, Report UIUCDCS-R-74-672, 1974.
  • [2] S. D. CONTE & C. de BOOR, Elementary Numerical Analysis: An Algorithmic Approach, McGraw-Hill, New York, 1972. MR 0202267 (34:2140)
  • [3] B. L. EHLE, On Padé Approximations to the Exponential Function and A-stable methods for the Numerical Solution of Initial Value Problems, Res. Rep. CSRR 2010, Dept. of Appl. Anal. and Comput. Sci., Univ. of Waterloo, Canada, 1969.
  • [4] W. H. ENRIGHT, "Second derivative multistep methods for stiff ordinary differential equations," SIAM J. Numer. Anal., v. 11, 1974, pp. 321-331. MR 50 # 3574. MR 0351083 (50:3574)
  • [5] C. W. GEAR, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 47 # 4447. MR 0315898 (47:4447)
  • [6] E. GRIEPENTROG, "Mehrschrittverfahren zur numerischen Integration von gewöhnlichen Differentialgleichungssystemen und asymptotische Exaktheit," Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe, v. 19, 1970, pp. 637-653. MR 47 # 9833. MR 0321300 (47:9833)
  • [7] R. JELTSCH, "Stiff stability and its relation to $ {A_0}$- and $ A(0)$-stability," SIAM J. Numer. Anal., v. 19, 1976, pp. 8-17. MR 0411174 (53:14913)
  • [8] R. JELTSCH, "Multistep methods using higher derivatives and damping at infinity," Math. Comp. (To appear.) MR 0428716 (55:1736)
  • [9] R. JELTSCH, "Multistep multiderivative methods and Hermite Birkhoff interpolation," Proc. 5th Manitoba Conference on Numerical Mathematics and Computing, October 1-4, 1975, Winnipeg. MR 0436600 (55:9543)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L05

Retrieve articles in all journals with MSC: 65L05

Additional Information

Keywords: Linear k-step methods using higher derivatives, behavior of the region of absolute stability at the origin, necessary condition for A-stability, Hermite interpolatory multistep multiderivative methods, Adams-type multistep multiderivative methods
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society