On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations

The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix.

A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence.

The order of convergence is determined in accordance with the smoothness scale.

Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semi-linear settings.

The convergence properties of the scheme are examined in relation to the regularity of the solution.