The Crank-Nicolson/Explicit Scheme for the Natural Convection Equations with Nonsmooth Initial Data

Abstract

In this article, a Crank-Nicolson/Explicit scheme is designed and analyzed for the time-dependent natural convection problem with nonsmooth initial data. The Galerkin finite element method (FEM) with stable MINI element is used for the velocity and pressure and linear polynomial for the temperature. The time discretization is based on the Crank-Nicolson scheme. In order to simplify the computations, the nonlinear terms are treated by the explicit scheme. The advantages of our numerical scheme can be list as follows: (1) The original problem is split into two linear subproblems, these subproblems can be solved in each time level in parallel and the computational sizes are smaller than the origin one. (2) A constant coefficient linear discrete algebraic system is obtained in each subproblem and the computation becomes easy. The main contributions of this work are the stability and convergence results of numerical solutions with nonsmooth initial data. Finally, some numerical results are presented to verify the established theoretical results and show the performances of the developed numerical scheme.