Abstract
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
Similar content being viewed by others
References
K. Burrage J.C. Butcher (1980) ArticleTitleNon-linear stability of a general class of differential equation methods BIT 20 185–203 Occurrence Handle583033 Occurrence Handle0431.65051
J.C. Butcher (1966) ArticleTitleOn the convergence of numerical methods for ordinary differential equations Math. Comp. 20 1–10 Occurrence Handle0141.13504 Occurrence Handle189251
J.C. Butcher (2001) ArticleTitleGeneral linear methods for stiff differential equations BIT 41 240–264 Occurrence Handle0983.65085 Occurrence Handle1837396
J.C. Butcher W. Wright (2003) ArticleTitleThe construction of practical general linear methods BIT 43 695–721 Occurrence Handle2068266 Occurrence Handle10.1023/B:BITN.0000009952.71388.23 Occurrence Handle1046.65054
C.F. Curtiss J.O. Hirschfelder (1952) ArticleTitleIntegration of stiff equations Proc. Nat. Acad. Sci. 38 235–243 Occurrence Handle47404 Occurrence Handle10.1073/pnas.38.3.235 Occurrence Handle0046.13602
E. Hairer G. Wanner (1991) Solving Ordinary Differential Equations II: Stiff and Differential-algebraic Problems Springer-Verlag Berlin Occurrence Handle0729.65051
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Butcher, J.C. High Order A-stable Numerical Methods for Stiff Problems. J Sci Comput 25, 51–66 (2005). https://doi.org/10.1007/s10915-004-4632-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10915-004-4632-8