Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Exponential fitting of matricial multistep methods for ordinary differential equations


Authors: E. F. Sarkany and W. Liniger
Journal: Math. Comp. 28 (1974), 1035-1052
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1974-0368441-1
MathSciNet review: 0368441
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of explicit or implicit multistep integration formulas for solving $ N \times N$ systems of ordinary differential equations. The coefficients of these formulas are diagonal matrices of order N, depending on a diagonal matrix of parameters Q of the same order. By definition, the formulas considered here are exact with respect to $ y' = - Dy + \phi (x,y)$ provided $ Q = hD,h$ is the integration step, and $ \phi $ belongs to a certain class of polynomials in the independent variable x. For arbitrary step number $ k \geqslant 1$, the coefficients of the formulas are given explicitly as functions of Q. The present formulas are generalizations of the Adams methods $ (Q = 0)$ and of the backward differentiation formulas $ (Q = + \infty )$. For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-h stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the Newton-Raphson method; it is well suited for stiff problems. The second implementation is a predictor-corrector approach. An error analysis is carried out for arbitrarily large Q. Finally, results of numerical test calculations are presented.


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

  • [1] E. F. SARKANY & W. E. BALL, "A predictor-corrector modification of the Cohen-Flatt-Certaine method for solving differential equations," Proceedings Joint Conference on Mathematical and Computer Aids to Design, Anaheim, Calif., 1969, p. 336.
  • [2] J. CERTAINE, "The solution of ordinary differential equations with large time constants," Mathematical Methods for Digital Computers, A. Ralston and H. S. Wilf (Editors), vol. 1, Wiley, New York, 1960, pp. 128-132. MR 22 #8691. MR 0117917 (22:8691)
  • [3] E. R. COHEN & H. P. FLATT, "Numerical solution of quasilinear equations," Codes for Reactor Computations, International Atomic Energy Agency, Vienna, 1961, pp. 461-484.
  • [4] K. G. GUDERLEY & C. C. HSU, "A predictor-corrector method for a certain class of stiff differential equations," Math. Comp., v. 26, 1972, pp. 51-69. MR 45 #8001. MR 0298952 (45:8001)
  • [5] W. LINIGER, "Global accuracy and A-stability of one- and two-step integration formulae for stiff ordinary differential equations," Conference on the Numerical Solution of Differential Equations (Dundee, 1969), Lecture Notes in Math., vol. 109, Springer-Verlag, Berlin, 1969, pp. 188-193.
  • [6] C. F. CURTISS & J. O. HIRSCHFELDER, "Integration of stiff equations," Proc. Nat. Acad. Sci. U.S.A., v. 38, 1952, pp. 235-243. MR 13, 873. MR 0047404 (13:873c)
  • [7] G. G. DAHLQUIST, "A special stability criterion for linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 3, 1963, pp. 27-43. MR 30 #715. MR 0170477 (30:715)
  • [8] O. B. WIDLUND, "A note on unconditionally stable linear multistep methods," Nordisk Tidskr. Informationsbehandling (BIT), v. 7, 1967, pp. 65-70. MR 35 #6373. MR 0215533 (35:6373)
  • [9] C. W. GEAR, "The automatic integration of stiff ordinary differential equations (With discussion)," Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), vol. I: Mathematics, Software, North-Holland, Amsterdam, 1969, pp. 187-193. MR 41 #4808. MR 0260180 (41:4808)
  • [10] F. ODEH & W. LINIGER, "A note on unconditional fixed-h stability of linear multistep methods," Computing (Arch. Elektron. Rechnen), v. 7, 1971, pp. 240-253. MR 45 #8006. MR 0298957 (45:8006)
  • [11] G. G. DAHLQUIST, "Stability and error bounds in the numerical integration of ordinary differential equations," Kungl. Tekn. Högsk. Stockholm, No. 130, 1959, 87 pp. MR 21 #1706. MR 0102921 (21:1706)
  • [12] W. LINIGER & R. A. WILLOUGHBY, "Efficient integration methods for stiff systems of ordinary differential equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 47-66. MR 41 #4809. MR 0260181 (41:4809)
  • [13] W. LINIGER, "A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear ordinary differential equations," Comm. ACM, v. 14, 1971, pp. 600-601. MR 44 #7754. MR 0290574 (44:7754)
  • [14] R. K. JAIN, "Some A-stable methods for stiff ordinary differential equations," Math. Comp., v. 26, 1972, pp. 71-77. MR 46 #2869. MR 0303733 (46:2869)
  • [15] S. NØRSETT, "An A-stable modification of the Adams-Bashforth methods," Conference on Numerical Solution of Differential Equations (Dundee, 1969), Lecture Notes in Math., vol. 109, Springer-Verlag, Berlin, 1969, pp. 214-219. MR 42 #2673. MR 0267771 (42:2673)
  • [16] E. F. SARKANY & W. LINIGER, Exponential Fitting of Matricial Multistep Methods for Ordinary Differential Equations, IBM Report RC 4149, Dec. 8, 1972.
  • [17] P. HENRICI, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962, Chap. 5. MR 24 #B1772. MR 0135729 (24:B1772)
  • [18] L. G. KELLY, Handbook of Numerical Methods and Applications, Addison-Wesley, Reading, Mass., 1967, p. 39.
  • [19] G. PETIT BOIS, Tables of Indefinite Integrals, Dover, New York, 1961, p. 145. MR 23 #A256. MR 0122924 (23:A256)
  • [20] Op. cit., Ref. 17, p. 222.
  • [21] G. G. DAHLQUIST, "Convergence and stability in the numerical integration of ordinary differential equations," Math. Scand., v. 4, 1956, pp. 33-53. MR 18, 338. MR 0080998 (18:338d)
  • [22] W. LINIGER, "A criterion for A-stability of linear multistep integration formulae," Computing (Arch. Elektron. Rechnen), v. 3, 1968, pp. 280-285. MR 39 #1120. MR 0239763 (39:1120)
  • [23] F. R. GANTMAHER, The Theory of Matrices, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959, Vol. 2, Chap. 15. MR 16, 438; 21 #6372c. MR 0065520 (16:438l)
  • [24] W. LINIGER & F. ODEH, "A-stable accurate averaging of multistep methods for stiff differential equations," IBM J. Res. Develop., v. 16, 1972, pp. 335-348. MR 0345416 (49:10152)
  • [25] R. H. ALLEN & C. POTTLE, "Stable integration methods for electronic circuit analysis with widely separated time constants," Proceedings Sixth Allerton Conference on Circuit Design and System Theory, University of Illinois, 1967, pp. 534-543.
  • [26] Op. cit., Ref. 17, p. 255.
  • [27] L. LAPIDUS & J. H. SEINFELD, Numerical Solution of Ordinary Differential Equations, Math. in Sci. and Engineering, vol. 74, Academic Press, New York, 1971, p. 182. MR 43 #7073. MR 0281355 (43:7073)
  • [28] G. BJUREL, et al., Survey of Stiff Ordinary Differential Equations, Report NA 70.11, Roy. Inst. of Tech., Stockholm, Sweden.
  • [29] M. E. FOWLER & R. M. WARTEN, "A numerical integration technique for ordinary differential equations with widely separated eigenvalues," IBM J. Res. Develop., v. 11, 1967, pp. 537-543. MR 35 #7586. MR 0216757 (35:7586)

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/S0025-5718-1974-0368441-1
Keywords: Ordinary differential equations, matricial multistep methods, exponential fitting, unconditional fixed-h stability
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society