The stability of rational approximations of cosine functions on Hilbert spaces
An abstract initial value problem of second order in time of the following type \[ u^{\prime \prime}(t) =-Au(t), \;\;t\in \mathbb R, \;\;\;u(0)=u_0, \;\;u^{\prime}(0)=v_0 \] with densely defined closed linear operator \(A\) on some complex Hilbert space is investigated. The operator is supposed to generate a cosine function (the problem is well posed). So called rational methods for the time discretization of this problem are considered. The presented results extended previous results obtained for selfadjoint operators. Stability for the proposed rational methods is proved under some hypotheses and in the particular case for Runge-Kutta and Runge-Kutta-Nyström methods the conditions where the hypotheses for stability are fulfilled are stated.
second order in time evolutionary problems
scientific article; zbMATH DE number 5382927
scientific article