Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations

In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold ? to provide dimension reduction; an efficient proper orthogonal decomposition?Greedy sampling method for identification of optimal and numerically stable approximations ? to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method ? to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs ? to provide certainty in our predictions; and an offline?online computational decomposition strategy for our RB approximation and associated error bound ? to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional “complex” enclosure ? a square with a small rectangle cutout ? parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.