On error estimates of the projection method for the time-dependent natural convection problem: First order scheme

In this article, a projection method (or fractional step method) is proposed and analyzed for the time-dependent natural convection problem in two dimensions. Based on this method, the considered problem is decoupled into two subproblems: one is for the velocity and temperature, and the other is for the pressure. Firstly, a linear elliptic equation is treated for the temperature; Then, a linear elliptic equation is solved for the velocity; Finally, a Poisson equation is considered for the pressure. Stability is performed and the convergence results of the projection method are derived. Furthermore, a modified projection scheme which leads to some improved error estimates is also proposed. Finally, some numerical examples are presented to verify the established theoretical findings.