Abstract
Implicit and explicit Adams-like multistep formulas are derived for equations of the typeP(d/dt)y=f(t,y) whereP is a polynomial with constant coefficients and where ∣∂f/∂y∣ is considered small compared with the roots ofP. Such equations appear for instance in control theory. An analysis of the local truncation error is performed and some examples are discussed where a considerable gain of computation time is obtained compared with classical methods. Finally some extensions of this method are mentioned in order to treat more general systems of differential equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. H. Anderson, R. B. Ball, and J. R. Voss,A numerical method for solving control differential equations on digital computers, JACM 7 (1960), 61–68.
D. G. Bettis,Numerical integration of products of Fourier and ordinary polynomials, Num. Math. 14 (1970), 421–434.
G. Bjurel,Preliminary report on: Modified linear multistep methods for a class of stiff ordinary differential equations, Report NA-69.02 (1969) Dept. of Computer Science, Royal Inst. of Techn., Stockholm, Sweden.
G. Bjurel,Supplement to report on Modified linear multistep methods for a class of stiff ordinary differential equations, Report NA-71.43 (1971) Dept. of Computer Science, Royal Inst. of Techn., Stockholm, Sweden.
G. Bjurel, G. Dahlquist, B. Lindberg, S. Linde, and L. Odén,Survey of stiff ordinary differential equations, Report NA-70.11 (1970) Dept. of Computer Science, Royal Inst. of Techn., Stockholm, Sweden.
P. Brock and F. J. Murray,The use of exponential sums in step by step integration, MTAC 6 (1952), 63–78, 138–150.
J. Certaine,The solution of ordinary differential equations with large time constants, A. Ralston, H. S. Wilf (Eds.), Math. Methods for Digital Computers, J. Wiley, New York, vol. 1 (1960).
G. Dahlquist,Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53.
G. Dahlquist,A numerical method for some ordinary differential equations with large Lipschitz constants, IFIP Congress (1968) Supplement, 132–136.
P. J. Davis,Interpolation and approximation, Blaisdell Publishing Company, New York (1965).
M. E. Fowler,A new numerical method for simulation, Simulation, May (1965).
M. E. Fowler, R. M. Warten,A numerical integration technique for ordinary differential equations with widely separated eigenvalues, IBM J. Res. Develop. LL (1967), 537–543.
W. Gautschi,Numerical integration of ordinary differential equations based on trigonometric polynomials, Num. Math. 3 (1961), 381–397.
C. W. Gear,Numerical initial value problems in ordinary differential equations, Prentice-Hall series in automatic computation, 1971.
J. D. Lambert,Linear multistep methods with mildly varying coefficients, Report from Dept. of Math., Univ. of Dundee, Scotland (1967).
W. Liniger,Global accuracy and A-stability of one- and two-step integration formulae for stiff ordinary differential equations, IBM rep. RC 2396 (1969).
W. L. Miranker,Matricial difference schemes, Report from Hebrew University of Jerusalem, Israel (1969).
S. P. Nørsett,An A-stable modification of the Adams-Basforth methods, Conf. on Numerical Solution of Differential Equations, Dundee (1969).
L. Odén,An experimental and theoretical analysis of the SAPS-method for stiff ordinary differential equations, Report NA-71.28 (1971) Dept. of Computer Science, Royal Inst. of Techn., Stockholm, Sweden.
Author information
Authors and Affiliations
Additional information
This research was supported in part by the Swedish National Council for Scientific Research and the Research Institute of the Swedish National Defence.
Rights and permissions
About this article
Cite this article
Bjurel, G. Modified linear multistep methods for a class of stiff ordinary differential equations. BIT 12, 142–160 (1972). https://doi.org/10.1007/BF01932809
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01932809