Second order fully discrete and divergence free conserving scheme for time-dependent conduction–convection equations

This work presents a Crank–Nicolson extrapolation scheme for the two-dimensional time-dependent conduction–convection equations. Mixed finite element method is applied for the spatial approximation of the velocity, pressure and temperature. The time discretization is based on the Crank–Nicolson scheme for the linear term and semi-implicit scheme for the nonlinear term. Moreover, the stability analysis and error estimations are derived. Finally, numerical tests confirm the theoretical results of the presented method and show the efficient method conserves the property of divergence free of the original equations to some extent.