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Semiexplicit $ A$-stable Runge-Kutta methods

Authors: G. J. Cooper and A. Sayfy
Journal: Math. Comp. 33 (1979), 541-556
MSC: Primary 65L05
MathSciNet review: 521275
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Abstract: An $ s - 1$ stage semiexplicit Runge-Kutta method is represented by an $ s \times s$ real lower triangular matrix where the number of implicit stages is given by the number of nonzero diagonal elements. It is shown that the maximum order attainable is s when $ s \leqslant 5$. Necessary and sufficient conditions for A-stability are derived and it is shown that there must be $ s - 1$ implicit stages if the order is s and $ s \leqslant 5$. Examples are given for $ s \leqslant 4$ where all the nonzero diagonal elements are equal. Additional problems arise when $ s > 4$; but when $ s = 6$, an A-stable method of order 5 is obtained. This method has five nonzero diagonal elements, and these elements are equal. Finally, a six stage A-stable method of order six is given. Again, this method has five nonzero (and equal) diagonal elements.

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Article copyright: © Copyright 1979 American Mathematical Society

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