Decoupling technology for systems of evolutionary equations

Abstract

Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a new level in time. The decoupling method, a significant approach to simplifying the problem, is based on the decomposition of the problem's operator matrix. The approximate solution is constructed based on the linear composition of solutions to auxiliary problems. The paper investigates decoupling variants based on extracting the diagonal part of the operator matrix and the lower and upper triangular submatrices. The study introduces a new decomposition approach, which involves splitting the operator matrix into rows and columns. The composition stage utilizes various variants of splitting schemes, showcasing the versatility of the approach. In additive operator-difference schemes, we can distinguish explicit-implicit schemes, factorized schemes for two-component splitting, and regularized schemes for general multi-component splitting. The study of stability of two- and three-level decoupling composition schemes is carried out using the theory of stability (correctness) of operator-difference schemes for finite-dimensional Hilbert spaces. The theoretical results of the decoupling technique for systems of evolution equations are illustrated on a test two-dimensional problem for a coupled system of two diffusion equations with inhomogeneous self- and cross-diffusion coefficients.