Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Some necessary conditions for convergence of the GBDF methods


Author: Mohamed Bin Suleiman
Journal: Math. Comp. 60 (1993), 635-649
MSC: Primary 65L05; Secondary 65L20
DOI: https://doi.org/10.1090/S0025-5718-1993-1176717-5
MathSciNet review: 1176717
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Generalized Backward Differentiation methods for solving stiff higher-order ordinary differential equations are described. The convergence, zero stability and consistency of these methods are defined. Next, the zero stability and consistency conditions necessary for convergence are proven. The order for which the methods are zero stable is also determined.


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

  • [1] C. W. Cryer, On the instability of high order backward difference multistep methods, BIT 12 (1972), 17-25. MR 0311112 (46:10208)
  • [2] C. W. Gear, The stability of numerical methods for second-order ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), 188-197. MR 0468191 (57:8029)
  • [3] G. Hall and M. B. Suleiman, Stability of Adams-type formulae for second-order ordinary differential equations, IMA J. Numer. Anal. 1 (1981), 427-438. MR 641320 (83a:65073)
  • [4] P. Henrici, Discrete variable methods in ordinary differential equations, Wiley, New York, 1962. MR 0135729 (24:B1772)
  • [5] F. T. Krogh, A variable step variable order multistep method for the numerical solution of ordinary differential equations, Proc. IFIP Congr., in Inform. Process. 68 (1968), 194-199. MR 0261790 (41:6402)
  • [6] M. B. Suleiman, Convergence of the variable order and variable stepsize direct integration methods for the solution of the higher order ordinary differential equation, Pertanika 8 (1985), 59-66.
  • [7] M. B. Suleiman and C. W. Gear, Treating a stiff single second-order ODE directly, J. Comput. Appl. Math. 27 (1989), 331-348. MR 1026367 (90i:65138)
  • [8] M. B. Suleiman, Solving higher order ODEs directly by the direct integration method, Appl. Math. Comput. 33 (1989), 197-219. MR 1017056 (90i:65139)
  • [9] J. M. Varah, A comparison of some numerical methods for two-point boundary value problems, Math. Comp. 28 (1974), 743-755. MR 0373300 (51:9500)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L05, 65L20

Retrieve articles in all journals with MSC: 65L05, 65L20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1176717-5
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society