Proceedings of I. Vekua Institute of Applied Mathematics Vol. 54-55, 2004-2005 APPROXIMATE SOLUTION OF CAUCHY PROBLEM FOR ABSTRACT HYPERBOLIC EQUATION USING UNITARY GROUP APPROXIMATION METHOD Lomtadze T., Rogava J., Tsiklauri M. Iv.Javakhishvili Tbilisi State University, I. Vekua Institute of Applied Mathematics, Tbilisi State University, 2 University Str., 0186 Tbilisi, Georgia e.mail: jrogava@viam.hepi.edu.ge (Received:24.02.2005; revised:12.09.2005) Abstract In the present work, on the basis of rational approximation of the unitary group, there is constructed two layer semi-discrete scheme of the fourth order precision for the approximate solution of Cauchy problem for abstract hyperbolic equation. The stability of the scheme is investigated. There is obtained an explicit estimation for the error of the approximate solution. Key words and phrases: abstract hyperbolic equation, unitary group, semi-discrete scheme. AMS subject classification: 65M15. Introduction The main incentives to present investigations was the very interesting in our view paper of the authors Baker G. A., Dougalis V. A., da Serbin S. M. [1]. As it is known, the solution to Cauchy problem for an abstract hyperbolic equation can be given by means of sine and cosine operator functions, where square root from the main operator is included in the argument. Using this formula, the above-mentioned authors for the equally distanced values of the time variable construct the precise three-layer semi-discrete scheme, whose transition opera- tor is a cosine-operator function. From this scheme one can obtain three-layer semi-discrete schemes of the any order precision, with the transition operator, which represents a rational approximation of the cosine-operator function. It must be pointed out, that in the denominator and numerator of this rational approximation there participate only even powers. The approximation of this type excludes the necessity of taking the square root from the main operator, what is a very important factor in view of practical usage. We look at the idea developed in the work [1] from another point of view. Namely, we can construct a scheme with any order precision for the approx- imate solution of abstract hyperbolic equation (in case of Cauchy problem)