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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Marginal stability and stabilization in the numerical integration of ordinary differential equations
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by H. Brunner PDF
Math. Comp. 24 (1970), 635-646 Request permission


Strongly stable and consistent multistep methods with maximum order are subject to marginal (or weak) stability. In this paper we introduce modified multistep methods whose coefficients depend linearly on the stepsize $h$ and a parameter $L$ in such a way that the order of the original method is not decreased. By choosing $L$ in a suitable manner (depending essentially on ${f_y}(x,y)$ of the differential equation $y’ = f(x,y)$ and on the growth parameters of the multistep method), marginal stability can be eliminated.
  • Hermann Brunner, Stabilization of optimal difference operators, Z. Angew. Math. Phys. 18 (1967), 438–444 (English, with German summary). MR 218022, DOI 10.1007/BF01601284
  • Hermann Brunner, Stabilisierung optimaler Differenzenverfahren zur numerischen Integration gewöhnlicher Differentialgleichungen, Abhandlung zur Erlangung der Würde eines Doktors der Mathematik der Eidgenössischen Technischen Hochschule Zürich, Juris Druck + Verlag, Zürich, 1969 (German). Dissertation No. 4182. MR 0264850
  • Germund Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53. MR 80998, DOI 10.7146/math.scand.a-10454
  • Germund Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm 130 (1959), 87. MR 102921
  • Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
  • Peter Henrici, Error propagation for difference method, John Wiley & Sons, Inc., New York-London, 1963. MR 0154416
  • Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
  • T. E. Hull and A. C. R. Newbery, Corrector formulas for multi-step integration methods, J. Soc. Indust. Appl. Math. 10 (1962), 351–369. MR 152150
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Additional Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Math. Comp. 24 (1970), 635-646
  • MSC: Primary 65.61
  • DOI:
  • MathSciNet review: 0273821