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BIT Numerical Mathematics

, Volume 24, Issue 3, pp 394–396 | Cite as

Linear stability of stiff differential equation solvers

  • T. J. Ypma
Scientific Notes

Abstract

When a linear multistep method is used to solve a stiff differential equationy′(x)=f(y(x)), producing an approximationy n toy(x n ), it is preferable to approximate the valuey′(x n ) in subsequent formulae by a value which exactly satisfies the corrector equation used, rather than by the valuef(y n ). We prove that the resulting method is stable if the underlying corrector equation is absolutely stable, provided that the residuals obtained in solving successive nonlinear equations remain uniformly bounded.

Keywords

Differential Equation Computational Mathematic Nonlinear Equation Linear Stability Multistep Method 
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References

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    C. W. Gear and Y. Saad,Iterative solution of linear equations in ODE codes, SIAM J. Sci. Stat. Comp. 4 (1983), 583–601.Google Scholar
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    J. D. Lambert,Computational Methods in Ordinary Differential Equations, Wiley, New York (1973).Google Scholar
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    H. H. Robertson and J. Williams,Some properties of algorithms for stiff differential equations, JIMA 16 (1975), 23–34.Google Scholar
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    L. F. Shampine,Evaluation of implicit formulas for the solution of ODE's., BIT 19(1979), 495–502.Google Scholar
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    L. F. Shampine,Implementation of implicit formulas for the solution of ODE's., SIAM J. Sci. Stat. Comp. 1(1980), 103–118.Google Scholar

Copyright information

© BIT Foundations 1984

Authors and Affiliations

  • T. J. Ypma
    • 1
  1. 1.Department of Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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