Semiexplicit 𝐴-stable Runge-Kutta methods

Cooper, G. J.; Sayfy, A.

doi:10.1090/S0025-5718-1979-0521275-1

Semiexplicit $A$-stable Runge-Kutta methods
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by G. J. Cooper and A. Sayfy PDF

Math. Comp. 33 (1979), 541-556 Request permission

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.

References

J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963), 185–201. MR 0152129, DOI 10.1017/S1446788700027932
J. C. Butcher, On the implementation of implicit Runge-Kutta methods, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 3, 237–240. MR 488746, DOI 10.1007/bf01932265
G. J. Cooper and J. H. Verner, Some explicit Runge-Kutta methods of high order, SIAM J. Numer. Anal. 9 (1972), 389–405. MR 317546, DOI 10.1137/0709037
Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532

Semi-Explicit Runge-Kutta Methods

Syvert P. Nørsett, $C$-polynomials for rational approximation to the exponential function, Numer. Math. 25 (1975/76), no. 1, 39–56. MR 410189, DOI 10.1007/BF01419527