Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Implementing second-derivative multistep methods using the Nordsieck polynomial representation


Author: G. K. Gupta
Journal: Math. Comp. 32 (1978), 13-18
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1978-0478630-7
MathSciNet review: 0478630
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A polynomial representation for the second-derivative linear multistep methods for solving ordinary differential equations is presented. This representation leads to an implementation of the second-derivative methods using the Nordsieck polynomial representation. Possible advantages of such an implementation are then discussed.


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

  • [1] G. G. DAHLQUIST (1963), "A special stability problem for linear multistep methods," BIT, v. 3, pp. 27-43. MR 0170477 (30:715)
  • [2] W. H. ENRIGHT (1974a), "Second derivative multistep methods for stiff ordinary differential equations," SIAM J. Numer. Anal., v. 11, pp. 321-331. MR 0351083 (50:3574)
  • [3] W. H. ENRIGHT (1974b), "Optimal second derivative methods for stiff systems," in Stiff Differential Systems, R. A. Willoughby (Editor), Plenum Press, New York, pp. 95-109.
  • [4] W. H. ENRIGHT, T. E. HULL & B. LINDBERG (1975), "Comparing numerical methods for stiff systems of ODE:s," BIT, v. 15, pp. 10-48.
  • [5] C. W. GEAR (1971), Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J. MR 0315898 (47:4447)
  • [6] Y. GENIN(1974), "An algebraic approach to A-stable linear multistep-multiderivative integration formulas," BIT, v. 14, pp. 382-406. MR 0368438 (51:4679)
  • [7] G. K. GUPTA (1976), "Some new high-order multistep formulae for solving stiff equations," Math. Comp., v. 30, pp. 417-432. MR 0423812 (54:11786)
  • [8] R. JELTSCH (1975), Note on A-Stability of Multistep Multiderivative Methods, unpublished report. MR 0411173 (53:14912)
  • [9] J. D. LAMBERT (1973), Computation Methods in Ordinary Differential Equations, Wiley, New York. MR 0423815 (54:11789)
  • [10] J. D. LAMBERT & S. T. SIGURDSSON (1972), "Multistep methods with variable matrix coefficients," SIAM J. Numer. Anal., v. 9, pp. 715-733. MR 0317548 (47:6095)
  • [11] W. LINIGER & F. ODEH (1972), "A-stable, accurate averaging of multistep methods for stiff differential systems," IBM J. Res. Develop., v. 16, pp. 335-348. MR 0345416 (49:10152)
  • [12] W. LINIGER & R. A. WILLOUGHBY (1970), "Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal., v. 7, pp. 47-66. MR 0260181 (41:4809)
  • [13] M. MAKELA, O. NEVANLINNA & A. H. SIPILA (1974), "Exponentially fitted multistep methods by generalized Hermite Birkhoff interpolation," BIT, v. 14, pp. 437-451. MR 0411176 (53:14915)
  • [14] G. J. MAKINSON (1968), "Stable high order implicit methods for the numerical solution of systems of differential equations," Comput. J., v. 11, pp. 305-310. MR 0235737 (38:4040)
  • [15] A. NORDSIECK (1962), "On the numerical integration of ordinary differential equations," Math. Comp., v. 16, pp. 22-49. MR 0136519 (24:B2552)
  • [16] C. S. WALLACE & G. K. GUPTA (1973), "General linear multistep methods to solve ordinary differential equations," Austral. Comput. J., v.5, 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

DOI: https://doi.org/10.1090/S0025-5718-1978-0478630-7
Keywords: Linear multistep methods, stiff differential equations, multiderivative methods, numerical solutions of ordinary differential equations
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society