Implicit Schwarz domain decomposition method for a Rayleigh–Bénard problem

This paper introduces an implicit Schwarz domain decomposition (SDD) method for solving the Rayleigh–Bénard convection problem. The motivation behind this work is to develop an alternative algorithm that eliminates the stagnation error observed in the alternating Schwarz domain decomposition method while enhancing information transmission across subdomain interfaces. A key question explored is whether the implicit SDD method is comparable to the alternating approach and whether it provides a real improvement in terms of efficiency and accuracy. The study focuses on establishing a theoretical proof of convergence for the implicit SDD method combined with Legendre collocation, ensuring its suitability for the Rayleigh–Bénard problem. Additionally, efforts are made to optimize computational cost, making the algorithm more practical for large-scale simulations. Numerical validation is carried out to assess its performance and confirm its advantages. Unlike the alternating approach, which requires solving multiple systems per time step, the implicit method solves a single system, potentially improving convergence speed. Theoretical analysis and numerical experiments demonstrate that this approach effectively improves the accuracy and reduces computational effort, particularly in the case of asymmetric domain decompositions.