Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A class of $ A$-stable advanced multistep methods


Authors: Jack Williams and Frank de Hoog
Journal: Math. Comp. 28 (1974), 163-177
MSC: Primary 65L05
MathSciNet review: 0356519
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A class of A-stable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of y' and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementation of the fourth order method is described and the method applied to some stiff equations. Numerical comparisons are made with Gear's method.


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

  • [1] O. Axelsson, "A class of A-stable methods." Nordisk Tidskr. Informationsbehandling (BIT), v. 9, 1969, pp. 185-199. MR 40 #8266. MR 0255059 (40:8266)
  • [2] F. H. Chipman, "A-stable Runge-Kutta processes," Nordisk Tidskr. Informationsbehandling (BIT), v. 11, 1971, pp. 384-388. MR 0295582 (45:4648)
  • [3] G. Dahlquist, "A special stability problem for linear multistep methods," Nordisk Tidskr. Informationsbehandling, v. 3, 1963, pp. 27-43. MR 30 #715. MR 0170477 (30:715)
  • [4] G. Dahlquist et al., Survey of Stiff Ordinary Differential Equations, The Royal Institute of Technology, Stockholm, Report NA 70.11, 1970.
  • [5] J. W. Daniel, Non-Linear Equations Arising in Deferred Correction of Initial Value Problems, MRC Technical Report No. 818, 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 123-128. (Spanish) MR 40 #8269. MR 0255062 (40:8269)
  • [6] J. W. Daniel, V. Pereyra & L. L. Schumaker, Iterated Deferred Corrections for Initial Value Problems, MRC Technical Report #808, Madison, Wis., 1967; Also in Acta Ci. Venezolana, v. 19, 1968, pp. 128-135. (Spanish) MR 40 #8270. MR 0255063 (40:8270)
  • [7] B. L. Ehle, "High order A-stable methods for the numerical solution of systems of D.E.'s," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 276-278. MR 39 #1119. MR 0239762 (39:1119)
  • [8] C. W. Gear, The Automatic Integration of Stiff Ordinary Differential Equations, Proc. IFIP Congress, Supplement, Booklet A: 81-85, 1968. MR 0260180 (41:4808)
  • [9] C. W. Gear, "DIFSUB for solution of ordinary differential equations," Comm. ACM, v. 14, 1971, pp. 185-190.
  • [10] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0315898 (47:4447)
  • [11] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)
  • [12] B. L. Hulme, "Discrete Galerkin and related one-step methods for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 881-891. MR 0315899 (47:4448)
  • [13] F. T. Krogh, On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations, Jet Propulsion Laboratory, Pasadena, Calif., Tech. Mem. No. 217, 1970.
  • [14] L. Lapidus & J. H. Seinfeld, Numerical Solution of Ordinary Differential Equations, Math. in Sci. and Engineering, vol. 74, Academic Press, New York and London, 1971. MR 43 #7073. MR 0281355 (43:7073)
  • [15] A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965. MR 32 #8479. MR 0191070 (32:8479)
  • [16] J. B. Rosser, "A Runge-Kutta for all seasons," SIAM Rev., v. 9, 1967, pp. 417-452. MR 36 #2325. MR 0219242 (36:2325)
  • [17] L. F. Shampine & H. A. Watts, "Block implicit one-step methods," Math. Comp., v. 23, 1969, pp. 731-740. MR 41 #9445. MR 0264854 (41:9445)
  • [18] H. A. Watts & L. F. Shampine, "A-stable block implicit one-step methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 12, 1972, pp. 252-266. MR 0307483 (46:6603)
  • [19] H. A. Watts, A-Stable Block Implicit One-Step Methods, Sandia Laboratories Report SC-RR-71 0296.
  • [20] K. Wright, "Some relationships between implicit Runge-Kutta, collocation and Lanczos $ \tau $ methods, and their stability properties," Nordisk Tidskr. Informationsbehandling (BIT), v. 10, 1970, pp. 217-227. MR 42 #1345. MR 0266439 (42:1345)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L05

Retrieve articles in all journals with MSC: 65L05


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0356519-8
PII: S 0025-5718(1974)0356519-8
Keywords: A-stable, advanced multistep methods, stiff systems
Article copyright: © Copyright 1974 American Mathematical Society