  This work develops scientific computing techniques to further the exploration
of using boundary control alone to optimize mixing in Stokes flows. The
theoretical foundation including mathematical model and the optimality
conditions for solving the optimal control has been established by Hu and Wu in
a series of work. The scalar being mixed is purely advected by the flow and the
control is exerted tangentially on the domain boundary through the Navier slip
conditions. The control design is motivated by the physical observations that
the moving or rotating walls accelerate mixing.
  A gradient descent-based optimization algorithm is designed. A critical
problem is the computation of the Gateaux derivative or the gradient of the
cost functional. Two methods are proposed: one is based on the Variational
Formula (VF) and one utilizes Algorithmic Differentiation (AD). The convergence
of the algorithm is studied and various designs of boundary control using
cosine and sine functions with time segmentation are computed.
  The algorithm has a first order convergence rate and the VF method is more
efficient by taking only one third of the time as the AD method when the
dimension of control basis is large. The numerical implementations show that
the boundary control produces similar mixing results as internal mixings in the
existing literature. The mixing effect becomes better when more diverse basis
control functions and more time segmentation are utilized. It is shown that the
mixing decay rate in time follows power rules, approximately. The numerical
study in this work suggests that boundary control alone could be an effective
strategy for mixing in incompressible fluid flows.
