A Numerical study of second order time stepping methods for the boussinesq equations

The Boussinesq approximation is a way to solve nonisothermall flow such as natural convection prob-lems, without having to solve for the full compressible formulation of the Navier-Stokes equations.Natural convection is related with an increasing number of fields including oceanography, meteorol-ogy and geophysical context [1, 2, 4]. Moreover, because of its presence both in nature and engi-neering applications, it has highly taken attention from academic societies. The accurate and efficientnumerical solution of these flows are known to be the core of many applications.In this context, we aimed to study a new, efficient and accurate numerical scheme for a family ofsecond order time stepping methods for the Boussinesq system, by extending an earlier study of [3]for the Navier-Stokes equations (NSE) based on the pioneering work for [5, 6]. The main idea of thisstudy is to incorporate linearizations and stabilization terms such that the discrete curvature solutionsin velocity, temperature and pressure are proportional to this combination. The method requires thesolution of only one linear system per time step. We prove unconditional stability of the method andalso derive a priori error bound. The derived theoretical results are supported by several numericalexperiments which prove the efficiency and the accuracy of the method