A special class of explicit linear multistep methods as basic methods for the correction in the dominant space technique

Author:
Peter Alfeld

Journal:
Math. Comp. **33** (1979), 1195-1212

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1979-0537965-0

MathSciNet review:
537965

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Abstract: A class of explicit linear multistep methods is suggested as basic methods for the CDS schemes introduced in [3]. These schemes are designed for the numerical solution of certain stiff ordinary differential equations, and operate with dominant eigenvalues, and the corresponding eigenvectors, of the Jacobian. The motivation, and the stability analysis for CDS schemes assumes that the eigensystem is constant. Here methods are introduced that perform particularly well if the eigensystem is not constant. In a certain sense the methods introduced here can be considered explicit approximations to the well-known implicit backward-differentiation formulas used by Gear [6] for the stiff option of his o.d.e. solver.

**[1]**Peter Alfeld,*Inverse linear multistep methods for the numerical solution of initial value problems of ordinary differential equations*, Math. Comp.**33**(1979), no. 145, 111–124. MR**514813**, https://doi.org/10.1090/S0025-5718-1979-0514813-6**[2]**P. ALFELD,*Correction in the Dominant Space*:*A New Technique for the Numerical Solution of Certain Stiff Initial Value Problems*, Ph. D. Thesis, University of Dundee, 1977.**[3]**P. Alfeld and J. D. Lambert,*Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems*, Math. Comp.**31**(1977), no. 140, 922–938. MR**0519719**, https://doi.org/10.1090/S0025-5718-1977-0519719-2**[4]**F. BASHFORTH & J. C. ADAMS,*Theories of Capillary Action*, Cambridge Univ. Press, Cambridge, 1883.**[5]**C. F. Curtiss and J. O. Hirschfelder,*Integration of stiff equations*, Proc. Nat. Acad. Sci. U. S. A.**38**(1952), 235–243. MR**0047404****[6]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[7]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[8]**J. D. Lambert,*Computational methods in ordinary differential equations*, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR**0423815**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0537965-0

Keywords:
Ordinary differential equations,
numerical analysis,
correction in the dominant space,
separably stiff systems,
interprojection,
backward-differentiation formulas

Article copyright:
© Copyright 1979
American Mathematical Society