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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
DOI: https://doi.org/10.1090/S0025-5718-1979-0521275-1
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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  • [1] J. C. BUTCHER, "Coefficients for the study of Runge-Kutta integration processes," J. Austral. Math. Soc., v. 3, 1963, pp. 185-201. MR 0152129 (27:2109)
  • [2] J. C. BUTCHER, "On the implementation of implicit Runge-Kutta methods," BIT, v. 16, 1976, pp. 237-240. MR 0488746 (58:8263)
  • [3] G. J. COOPER & J. H. VERNER, "Some explicit Runge-Kutta methods of high order," SIAM J. Numer. Anal., v. 9, 1972, pp. 389-405. MR 0317546 (47:6093)
  • [4] G. G. DAHLQUIST, "A special stability problem for linear multistep methods," BIT, v. 3, 1963, pp. 27-43. MR 0170477 (30:715)
  • [5] S. P. NORSETT, Semi-Explicit Runge-Kutta Methods, Mathematics Department, University of Trondheim, Reprint No. 6/74.
  • [6] S. P. NORSETT, "C-polynomials for rational approximation to the exponential function," Numer. Math., v. 25, 1975, pp. 39-56. MR 0410189 (53:13939)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0521275-1
Article copyright: © Copyright 1979 American Mathematical Society

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