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Mathematics of Computation

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Marginal stability and stabilization in the numerical integration of ordinary differential equations

Author: H. Brunner
Journal: Math. Comp. 24 (1970), 635-646
MSC: Primary 65.61
MathSciNet review: 0273821
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Abstract: 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.

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

  • Hermann Brunner, Stabilization of optimal difference operators, Z. Angew. Math. Phys. 18 (1967), 438–444 (English, with German summary). MR 218022, DOI
  • Hermann Brunner, Stabilisierung optimaler Differenzenverfahren zur numerischen Integration gewöhnlicher Differentialgleichungen, Abhandlung zur Erlangung der Würde eines Doktors der Mathematik der Eidgenössischen Technischen Hochschule, Zürich. Dissertation No. 4182, Juris Druck + Verlag, Zürich, 1969 (German). MR 0264850
  • Germund Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53. MR 80998, DOI
  • 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 and 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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Keywords: Nonlinear ordinary differential equations of order one, numerical solution, optimal multistep methods, marginal stability, stabilization
Article copyright: © Copyright 1970 American Mathematical Society