Journal of Scientific Computing

, Volume 25, Issue 1–2, pp 51–66 | Cite as

High order A-stable numerical methods for stiff problems

  • J. C. Butcher


Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge-Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order.

Key words

Stiff differential equations A-stable methods general linear methods inherent RK stability 


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  1. 1.
    Burrage, K., and Butcher, J. C. (1980). Non-linear stability of a general class of differential equation methods.BIT 20, 185–203.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Butcher, J. C. (1966). On the convergence of numerical methods for ordinary differential equations.Math. Comp. 20, 1–10.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Butcher, J. C. (2001). General linear methods for stiff differential equations.BIT 41, 240–264.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Butcher, J. C., and Wright, W. (2003). The construction of practical general linear methods,BIT 43, 695–721.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Curtiss, C. F., and Hirschfelder, J. O. (1952). Integration of stiff equations.Proc. Nat. Acad. Sci. 38, 235–243.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hairer, E., and Wanner, G. (1991).Solving Ordinary Differential Equations II: Stiff and Differential-algebraic Problems. Springer-Verlag, Berlin.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc 2005

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

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