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abstract differential equations; semigroups of operators; rational approximations; $A$-stability
The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.
[1] N.  Bourbaki: Fonctions d’une variable réelle (théorie élémentaire). Hermann & Cie, Paris, 1961. (French) Zbl 0131.05001
[2] J. C.  Butcher: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley & Sons, Chichester, 1987. MR 0878564 | Zbl 0616.65072
[3] N.  Dunford, J.  Schwartz: Linear Operators, Vol.  I. Interscience, New York-London, 1963. MR 0188745
[4] T.  Kato: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York, 1966. MR 0203473 | Zbl 0148.12601
[5] M. Práger, J.  Taufer, E.  Vitásek: Overimplicit multistep methods. Apl. Mat. 18 (1973), 399–421. MR 0366041
[6] K.  Yosida: Functional analysis. Springer-Verlag, Berlin-Heidelberg-New York, 1971. Zbl 0217.16001
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