Decomposition-composition schemes for systems of second-order evolutionary equations

We consider numerical methods for approximate solution of the Cauchy problem for coupled systems of second-order evolution equations. Simplification of the problem at a new layer in time is achieved by separating simpler subproblems for the individual components of the solution. The computational technique of decomposition-composition consists of two steps. First, the decomposition of the operator matrix of the problem is performed, and then an approximate solution is constructed based on a linear composition of the solutions of the auxiliary problems. In this paper, we investigate decomposition variants based on the extraction of the diagonal part, lower and upper triangular submatrices of the operator matrix, as well as when splitting the operator matrix into rows and columns. Different variants of splitting schemes are used at the composition stage. In two-component decomposition, explicit-implicit schemes and factorized schemes are distinguished. Regularized additive schemes are used in multi-component splitting. The study of stability of three-layer decomposition-composition schemes is carried out on the basis of stability theory of operator-difference schemes in finite-dimensional Hilbert spaces.