REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

Abstract

In this paper we present reduced basis 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 POD-Greedy sampling method for identification of optimal and numerically stable approximationsto 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 reduced basis approximation and associated outputs -to provide certainty in our predictions; and an Offline-Online computational decomposition strategy for our reduced basis 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 cut-out -parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the reduced basis approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…[154] for steady incompressible Navier-Stokes equations). However, the most important challenge deals with the reliability and/or the certification of the methodology in the unsteady parabolic problems [113,77,81]. In such cases, the exponential growth of the estimate seriously compromises a priori and a posteriori error estimates, yielding bounds which are limited to modest (final) times and modest Reynolds numbers [73].…”

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

“…These approaches have been proposed in [26,48] and more recently in a natural norm framework [9], focusing on physical parameters (Reynolds, Prandtl, Grashof numbers). Further developments have combined physical and geometrical parameters [11,46], dealing also with time-dependent Boussinesq equations [46,19].…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…Certified RB methods have been applied to parametrized (moderate Reynolds) unsteady viscous flows in [70], where a nonisothermal viscous flow is modelled by Boussinesq equations describing natural convection. Parameters are the Grashof number and the gravity direction.…”

“…Moreover, we provide detailed remarks and references about extensions of these techniques and alternative strategies. We do not address in this review the case of combined time and parameter-dependent problems; the interested reader can refer to some recent works concerning error estimates for ROMs in the case of acoustic Helmholtz and incompressible Navier-Stokes equations [63], the Boussinesq equations [70], and the viscous Burgers' equation using the method of lines [93] or in the space-time formulation [131].…”