The decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with nonsmooth initial data

Abstract

In this paper, the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations is considered with nonsmooth initial data. Our numerical scheme is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms for the temporal discretization, standard Galerkin finite element method is used to the spatial discretization. In order to improve the computational efficiency, the decoupled method is introduced, as a consequence the original problem is split into two linear subproblems, and these subproblems can be solved in parallel. We verify that our numerical scheme is almost unconditionally stable for the nonsmooth initial data (u0, θ0) with the divergence-free condition. Furthermore, under some stability conditions, we show that the error estimates for velocity and temperature in L2 norm is of the order O(h2+Δt32), in H1 norm is of the order O(h2+Δt), and the error estimate for pressure in a certain norm is of the order O(h2+Δt). Finally, some numerical examples are provided to verify the established theoretical findings and test the performances of the developed numerical scheme.