The Stability And Convergence For Boussinesq Equations With Smooth And Non-smooth Initial Data

As is known,many practical problems in science and engineering can be described by linear or nonlinear developmental differential equation or equations.Because people have limited understanding of the nature of nonlinear phenomena,numerical simulation is an important method to study the properties and properties of these equations.For the nonlinear partial differential equations,the smoothness of the initial solution has important influence on the stability and convergence of long time solution.Based on the smooth and non-smooth initial data,the stability and convergence of the first and second order numerical schemes of Boussinesq equations are analyzed.Firstly,the H2-stability of the first order fully discrete Galerkin finite element methods for the Boussinesq equations with smooth and non-smooth initial data is con-sidered.The finite element spatial discretization for the Boussinesq equations is based on the mixed finite element method,and the temporal treatments of the spatial dis-crete Boussinesq equations include the implicit scheme,the semi-implicit scheme,the implicit/explicit scheme and the explicit scheme.The H2-stability results of the above numerical schemes are established.First of all,we prove that the implicit and semi-implicit schemes are the H2-unconditional stable.In the next place,we show that the impliciu/explicit scheme is H2-almost unconditional stable with the initial data belong to H1 and H2,and the similar results are obtained for the semi-implicit/explicit scheme in the case of the initial data belong to L2.Then we show that the explicit scheme is the H2-conditional stable.Eventually,some numerical examples are provided to verify the established theoretical findings and confirm the corresponding H2 stability analysis of the different numerical schemes.Secondly,the stability and convergence of the decoupled Crank-Nicolson/Adams-Bashforth scheme for the Boussinesq equations with smooth initial data is considered,together with a stable mixed finite element spatial discretization.The temporal treat?ment of the spatial discrete Boussinesq equations is based on the implicit Crank-Nicolson scheme for the linear terms and the explicit Adams-Bashforth scheme for the nonlinear terms.Thanks to the decoupled scheme,the considered problem is decoupled into two subproblems,and these subproblems can be solved in parallel.We prove a restriction on the timestep that guarantees stability and obtains optimal error estimates of numerical solutions,and provide several numerical experiments that test the performances of the developed numerical scheme and verify the established theoretical findings.Finally,the stability and convergence of the decoupled Crank-Nicolson/Adams-Bashforth scheme for the Boussinesq equations is considered with nonsmooth initial data.To begin with,we verify that our numerical scheme is almost unconditionally stable.The next,under some stability conditions,we show that the error estimates for velocity and temperature in L2 norm is of the order O(h2 +?t3/2),in H1 norm is of the order O(h2 + ?t),and the error estimate for pressure in a certain norm is of the order O(h2 + ?t).Lastly,some numerical examples are provided to verify the validity of the numerical scheme.