Abstract <p>Numerical methods for approximate solution of the Cauchy problem for coupled systems of second-order evolution equations are considered. The problem at a new time level is simplified by considering simpler subproblems for individual components of the solution. The decomposition–composition technique consists of two stages. First, the operator matrix of the problem is decomposed. Then an approximate solution is constructed using a linear composition of solutions to auxiliary problems. We investigate decomposition variants based on extracting the diagonal part of the operator matrix and its lower and upper triangular submatrices and on splitting the operator matrix into rows and columns. Various splitting schemes are used at the composition stage. In the case of two-component splitting, we can apply explicit–implicit or factorized schemes. Regularized additive schemes are used for multicomponent splitting. The stability of three-level decomposition–composition schemes is analyzed by applying the theory of stability (well-posedness) of operator-difference schemes in finite-dimensional Hilbert spaces.</p>