Abstract
Strongly and weakly stable linear multistep methods can behave very differently. The latter class can produce spurious oscillations in some of the cases for which the former class works flawlessly. The main question is if we can find a well defined property which clearly tells the difference between them. There are many explanations from different viewpoints. We cite Spijker’s example which shows that the explicit two step midpoint method is unstable with respect to the Spijker norm. We show that this result can be extended for the general weakly stable case.
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References
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
Faragó, I., Mincsovics, M.E., Fekete, I.: Notes on the basic notions in nonlinear numerical analysis. Electron. J. Qual. Theory Diff. Equat. 6, 1–22 (2012). Proc. 9th Coll. Qualitative Theory of Diff. Equ. 2011
Mincsovics, M.E.: Stability of one-step and linear multistep methods - a matrix technique approach. Electron. J. Qual. Theory Diff. Equat. 15, 1–10 (2016). https://doi.org/10.14232/ejqtde.2016.8.15. Proc. 10th Coll. Qualitative Theory of Diff. Equ. 2015
Mincsovics, M.E.: Note on the stability of strongly stable linear multistep methods. In: AIP Conference Proceedings, vol. 1895, no. 1, p. 110006 (2017). https://doi.org/10.1063/1.5007412
Spijker, M.N.: Stability and convergence of finite-difference methods. Thesis, University of Leiden (1968)
Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer, Heidelberg (1973). https://doi.org/10.1007/978-3-642-65471-8
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Mincsovics, M.E. (2019). Spijker’s Example and Its Extension. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_40
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DOI: https://doi.org/10.1007/978-3-030-11539-5_40
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