The optimal control of thermally convective flows is usually modeled by an
optimization problem with constraints of Boussinesq equations that consist of
the Navier-Stokes equation and an advection-diffusion equation. This optimal
control problem is challenging from both theoretical analysis and algorithmic
design perspectives. For example, the nonlinearity and coupling of fluid flows
and energy transports prevent direct applications of gradient type algorithms
in practice. In this paper, we propose an efficient numerical method to solve
this problem based on the operator splitting and optimization techniques. In
particular, we employ the Marchuk-Yanenko method leveraged by the
L2βprojection for the time discretization of the Boussinesq equations so
that the Boussinesq equations are decomposed into some easier linear equations
without any difficulty in deriving the corresponding adjoint system.
Consequently, at each iteration, four easy linear advection-diffusion equations
and two degenerated Stokes equations at each time step are needed to be solved
for computing a gradient. Then, we apply the Bercovier-Pironneau finite element
method for space discretization, and design a BFGS type algorithm for solving
the fully discretized optimal control problem. We look into the structure of
the problem, and design a meticulous strategy to seek step sizes for the BFGS
efficiently. Efficiency of the numerical approach is promisingly validated by
the results of some preliminary numerical experiments