An artificial compressibility SAV finite element method for the time-dependent natural convection problem

Abstract

In this article, we present, analyze, and test a first-order implicit-explicit type scheme based on the artificial compressibility (AC) method and the scalar auxiliary variable (SAV) approach for solving the time-dependent natural convection problem. The proposed method is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization, and explicit treatment for nonlinear terms. It effectively decouples the velocity and pressure via artificial compressibility, thereby reducing computational complexity and execution time. Moreover, we use the SAV approach for the convective terms, it leads to the scheme is linear, and only require solving a sequence of linear differential equation with constant coefficients at each time step. It is proved that the solution of this method is bounded and satisfying the law of energy dissipation. The rigorous error estimate of velocity, pressure, and temperature are derived. Finally, some numerical tests are implemented to verify the theoretical analysis and illustrate the efficiency of the presented method.