Abstract
This paper considers the nonlinear stability of general linear methods. The diagonal matrix of an algebraically stable method is shown to be fixed. Readily testable necessary and sufficient conditions are obtained for algebraic stability and, more generally, for nonlinear stability in closed disk regions of the complex plane. It is shown that the latter criteria are satisfied by some explicit methods. It is also shown that certain methods, including some that are L-stable, suffer from nonautonomous instability along the negative real line near zero. A loose classification of methods is given according to nonlinear stability properties.
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Hill, A.T. Nonlinear Stability of General Linear Methods. Numer. Math. 103, 611–629 (2006). https://doi.org/10.1007/s00211-006-0011-8
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DOI: https://doi.org/10.1007/s00211-006-0011-8