Operator-Difference Schemes for a Class of Systems of Evolution Equations

For a special system of evolution equations of first order, discrete time approximations for the approximate solution of the Cauchy problem are considered. Such problems arise after the spatial approximation in the Schrödinger equation and the subsequent separation of the imaginary and real parts and in nonstationary problems of acoustics and electrodynamics. Unconditionally stable two-time-level operator-difference weighted schemes are constructed. The second class of difference schemes is based on the formal passage to explicit operator-difference schemes for evolution equations of second order when explicit-implicit approximation is used for isolated equations of the system. The regularization of such schemes in order to obtain unconditionally stable operator-difference schemes is discussed. Splitting schemes involving the solution of simplest problems at each time step are constructed.