Stability ordinates of Adams predictor-corrector methods
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How far the stability domain of a numerical method for approximating solutions to differential equations extends along the imaginary axis indicates how useful the method is for approximating solutions to wave equations; this maximum extent is termed the imaginary stability boundary, also known as the stability ordinate. It has previously been shown that exactly half of Adams-Bashforth (AB), Adams-Moulton (AM), and staggered Adams-Bashforth methods have nonzero stability ordinates. In this paper, we consider two categories of Adams predictor-corrector methods and prove that they follow a similar pattern. In particular, if \(p\) is the order of the method, AB\(p\)-AM\(p\) methods have nonzero stability ordinate only for \(p = 1, 2, \ 5, 6,\ 9, 10, \ldots \), and AB(\(p-\)1)-AM\(p\) methods have nonzero stability ordinates only for \(p = 3, 4, \ 7, 8, \ 11, 12, \ldots \).
KeywordsAdams methods Predictor-corrector Imaginary stability boundary Linear multistep methods Finite difference methods Stability region Stability ordinate
Mathematics Subject Classification65L06 65L12 65L20 65M06 65M12
The authors are extremely grateful to Ernst Hairer for suggesting major simplifications in a previous form of this manuscript, in particular with regard to using the backward difference forms of AB and AM methods. We are also grateful for helpful comments from the referees that allowed us to improve our manuscript.
- 3.Ghrist, M: High-order Finite Difference Methods for Wave Equations. Ph.D. thesis, Department of Applied Mathematics, University of Colorado-Boulder, Boulder, CO (2000)Google Scholar
- 9.Iserles, A.: Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)Google Scholar