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

Brunner, H.

doi:10.1090/S0025-5718-1970-0273821-5

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

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

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