2D Newton Schwarz Legendre Collocation Method for a Convection Problem

In this work, an alternate Schwarz domain decomposition method is proposed to solve a Rayleigh–Bénard problem. The problem is modeled with the incompressible Navier–Stokes equations coupled with a heat equation in a rectangular domain. The Boussinesq approximation is considered. The nonlinearity is solved with Newton’s method. Each iteration of Newton’s method is discretized with an alternating Schwarz scheme, and each Schwarz problem is solved with a Legendre collocation method. The original domain is divided into several subdomains in both directions of the plane. Legendre collocation meshes are coarse, so the problem in each subdomain is well conditioned, and the size of the total mesh can grow by increasing the number of subdomains. In this way, the ill conditioning of Legendre collocation is overcome. The present work achieves an efficient alternating Schwarz algorithm such that the number of subdomains can be increased indefinitely in both directions of the plane. The method has been validated with a benchmark with numerical solutions obtained with other methods and with real experiments. Thanks to this domain decomposition method, the aspect ratio and Rayleigh number can be increased considerably by adding subdomains. Rayleigh values near to the turbulent regime can be reached. Namely, the great advantage of this method is that we obtain solutions close to turbulence, or in domains with large aspect ratios, by solving systems of linear equations with well-conditioned matrices of maximum size one thousand. This is an advantage over other methods that require solving systems with huge matrices of the order of several million, usually with conditioning problems. The computational cost is comparable to other methods, and the code is parallelizable.