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A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear systems of ordinary differential equations

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W. LinigerAuthors Info & Claims

Communications of the ACM, Volume 14, Issue 9

Pages 600 - 601

https://doi.org/10.1145/362663.362745

Published: 01 September 1971 Publication History

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Abstract

In the numerical solution of ordinary differential equations, certain implicit linear multistep formulas, i.e. formulas of type ∑^k_j=0 α_jx_n+j - h ∑^k_j=0 β_jx_n+j = 0, (1) with β_k> ≠ 0, have long been favored because they exhibit strong (fixed-h) stability. Lately, it has been observed [1-3] that some special methods of this type are unconditionally fixed-h stable with respect to the step size. This property is of great importance for the efficient solution of stiff [4] systems of differential equations, i.e. systems with widely separated time constants. Such special methods make it possible to integrate stiff systems using a step size which is large relative to the rate of change of the fast-varying components of the solution.

References

[1]

Dahlquist, G.G. special stability criuterion for linear multistep methods, B1T3 (1963), 22-43.

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[2]

Gear, C.W. The automatic integration of stiff ordinary differential equations. Information Processing 1968, North-Holland Pub. Co., Amsterdam, pp. 187-193.

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[3]

Liniger, W., and Willoughby, R.A. Efficient integration methods for stiff systems of ordinary differential equations. SIAM J. Numer. Anal. 7 (1970), 47-66.

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[4]

Curtiss, C.F., and Hirschfelder, J.O. Integration of stiff equations. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 62-78, 138-150.

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Cited By

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Xu JHuang YQu Z(2020)An efficient and unconditionally stable numerical algorithm for nonlinear structural dynamicsInternational Journal for Numerical Methods in Engineering10.1002/nme.6456121:20(4614-4629)Online publication date: 21-Jul-2020
https://doi.org/10.1002/nme.6456
BEAM RWARMING R(2013)An implicit factored scheme for the compressible Navier-Stokes equations. II - The numerical ODE connection4th Computational Fluid Dynamics Conference10.2514/6.1979-1446Online publication date: 18-Feb-2013
https://doi.org/10.2514/6.1979-1446
Brambilla AMaffezzoni P(2003)Envelope-following method to compute steady-state solutions of electrical circuitsIEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications10.1109/TCSI.2003.80889750:3(407-417)Online publication date: Mar-2003
https://doi.org/10.1109/TCSI.2003.808897
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A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear systems of ordinary differential equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

Communications of the ACM Volume 14, Issue 9

Sept. 1971

36 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/362663

Editor:
M. Stuart Lynn
IBM Scientific Center, Houston, TX

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Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 September 1971

Published in CACM Volume 14, Issue 9

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Xu JHuang YQu Z(2020)An efficient and unconditionally stable numerical algorithm for nonlinear structural dynamicsInternational Journal for Numerical Methods in Engineering10.1002/nme.6456121:20(4614-4629)Online publication date: 21-Jul-2020
https://doi.org/10.1002/nme.6456
BEAM RWARMING R(2013)An implicit factored scheme for the compressible Navier-Stokes equations. II - The numerical ODE connection4th Computational Fluid Dynamics Conference10.2514/6.1979-1446Online publication date: 18-Feb-2013
https://doi.org/10.2514/6.1979-1446
Brambilla AMaffezzoni P(2003)Envelope-following method to compute steady-state solutions of electrical circuitsIEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications10.1109/TCSI.2003.80889750:3(407-417)Online publication date: Mar-2003
https://doi.org/10.1109/TCSI.2003.808897
Spijker M(1996)Stiffness in numerical initial-value problemsJournal of Computational and Applied Mathematics10.1016/0377-0427(96)00009-X72:2(393-406)Online publication date: 13-Aug-1996
https://dl.acm.org/doi/10.1016/0377-0427%2896%2900009-X
Spijker M(1995)The effect of the stopping of the Newton iteration in implicit linear multistep methodsApplied Numerical Mathematics10.1016/0168-9274(95)00064-218:1-3(367-386)Online publication date: 1-Sep-1995
https://dl.acm.org/doi/10.1016/0168-9274%2895%2900064-2
Jackson KKværnø ANørsett S(1994)The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulasApplied Numerical Mathematics10.1016/0168-9274(94)00031-X15:3(341-356)Online publication date: 1-Oct-1994
https://dl.acm.org/doi/10.1016/0168-9274%2894%2900031-X
Sugiura HTorii T(1991)A method for constructing generalized Runge-Kutta methodsJournal of Computational and Applied Mathematics10.1016/0377-0427(91)90185-M38:1-3(399-410)Online publication date: 23-Dec-1991
https://dl.acm.org/doi/10.1016/0377-0427%2891%2990185-M
Zavorin A(1979)Stable iterative realization of some implicit methods of integrationUSSR Computational Mathematics and Mathematical Physics10.1016/0041-5553(79)90072-719:1(125-132)Online publication date: Jan-1979
https://doi.org/10.1016/0041-5553(79)90072-7
Sarkany ELiniger W(1974)Exponential fitting of matricial multistep methods for ordinary differential equationsMathematics of Computation10.1090/S0025-5718-1974-0368441-128:128(1035-1052)Online publication date: 1974
https://doi.org/10.1090/S0025-5718-1974-0368441-1
Oesterhelt G(1974)Mehrschrittverfahren zur numerischen Integration von Differentialgleichungssystemen mit stark verschiedenen ZeitkonstantenMultistep methods for the numerical integration of systems of differential equations with widely separated time constantsComputing10.1007/BF0224172113:3-4(279-298)Online publication date: Sep-1974
https://doi.org/10.1007/BF02241721
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Linearization techniques for singular initial-value problems of ordinary differential equations

Numerical integration of ordinary differential equations with rapidly oscillatory factors

Approximate solution of linear ordinary differential equations with variable coefficients

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