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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of explicit or implicit multistep integration formulas for solving 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 provided is the integration step, and belongs to a certain class of polynomials in the independent variable *x*. For arbitrary step number , the coefficients of the formulas are given explicitly as functions of *Q*. The present formulas are generalizations of the Adams methods and of the backward differentiation formulas . 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.

**[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, Wiley, New York, 1960, pp. 128–132. MR**0117917****[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]**Karl G. Guderley and Chen-chi Hsu,*A predictor-corrector method for a certain class of stiff differential equations*, Math. Comp.**26**(1972), 51–69. MR**0298952**, https://doi.org/10.1090/S0025-5718-1972-0298952-7**[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 and J. O. Hirschfelder,*Integration of stiff equations*, Proc. Nat. Acad. Sci. U. S. A.**38**(1952), 235–243. MR**0047404****[7]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[8]**Olof B. Widlund,*A note on unconditionally stable linear multistep methods*, Nordisk Tidskr. Informations-Behandling**7**(1967), 65–70. MR**0215533****[9]**C. W. Gear,*The automatic integration of stiff ordinary differential equations.*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 187–193. MR**0260180****[10]**F. Odeh and W. Liniger,*A note on unconditional fixed-ℎ stability of linear multistep formulae*, Computing (Arch. Elektron. Rechnen)**7**(1971), 240–253 (English, with German summary). MR**0298957****[11]**Germund Dahlquist,*Stability and error bounds in the numerical integration of ordinary differential equations*, Kungl. Tekn. Högsk. Handl. Stockholm. No.**130**(1959), 87. MR**0102921****[12]**Werner Liniger and Ralph A. Willoughby,*Efficient integration methods for stiff systems of ordinary differential equations*, SIAM J. Numer. Anal.**7**(1970), 47–66. MR**0260181**, https://doi.org/10.1137/0707002**[13]**W. Liniger,*A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear systems of ordinary differential equations*, Comm. ACM**14**(1971), 600–601. MR**0290574**, https://doi.org/10.1145/362663.362745**[14]**R. K. Jain,*Some 𝐴-stable methods for stiff ordinary differential equations*, Math. Comp.**26**(1972), 71–77. MR**0303733**, https://doi.org/10.1090/S0025-5718-1972-0303733-1**[15]**Syvert P. Nørsett,*An 𝐴-stable modification of the Adams-Bashforth methods*, Conf. on Numerical Solution of Differential Equations (Dundee, 1969) Springer, Berlin, 1969, pp. 214–219. MR**0267771****[16]**E. F. SARKANY & W. LINIGER,*Exponential Fitting of Matricial Multistep Methods for Ordinary Differential Equations*, IBM Report RC 4149, Dec. 8, 1972.**[17]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[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 Publications, Inc., New York, 1961. MR**0122924****[20]**Op. cit., Ref. 17, p. 222.**[21]**Germund Dahlquist,*Convergence and stability in the numerical integration of ordinary differential equations*, Math. Scand.**4**(1956), 33–53. MR**0080998**, https://doi.org/10.7146/math.scand.a-10454**[22]**W. Liniger,*A criterion for 𝐴-stability of linear multistep integration formulae.*, Computing (Arch. Elektron. Rechnen)**3**(1968), 280–285 (English, with German summary). MR**0239763****[23]**F. R. Gantmaher,*Teoriya matric*, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (Russian). MR**0065520****[24]**W. Liniger and F. Odeh,*𝐴-stable, accurate averaging of multistep methods for stiff differential equations*, IBM J. Res. Develop.**16**(1972), 335–348. Mathematics of numerical computation. MR**0345416**, https://doi.org/10.1147/rd.164.0335**[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]**Leon Lapidus and John H. Seinfeld,*Numerical solution of ordinary differential equations*, Mathematics in Science and Engineering, Vol. 74, Academic Press, New York-London, 1971. MR**0281355****[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 and R. M. Warten,*A numerical integration technique for ordinary differential equations with widely separated eigenvalues*, IBM J. Res. Develop**11**(1967), 537–543. MR**0216757**, https://doi.org/10.1147/rd.115.0537

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