Accelerating Convergence of a Natural Convection Solver by Continuous Data Assimilation

The Picard iteration for the Boussinesq model of natural convection can be an attractive solver because it stably decouples the fluid equations from the temperature equation (for contrast, the Newton iteration does not stably decouple). However, the convergence of Picard for this system is at best linear, but slows as the Rayleigh number increases, eventually failing for even moderately sized Rayleigh numbers. We consider this solver in the setting where sparse solution data is available, e.g., from data measurements or solution observables, and enhance Picard by incorporating the partial solution data into the iteration using a continuous data assimilation (CDA) approach. We prove that our approach scales the linear convergence rate by $H^{\frac{1}{2}},$ with nudging velocity (or both velocity and temperature) producing the best results. This implies that when Picard is converging, CDA will accelerate convergence, and when Picard is not converging, CDA (with enough partial solution data) will enable convergence. In the case of noisy partial solution data, we prove that the linear convergence rate of the nonlinear residual is similarly scaled by $H^{\frac{1}{2}}$ but the accuracy is limited by the accuracy of the partial solution data. This is overcome by using CDA-Picard to achieve a specific tolerance for the residual and then switching the solver to Newton. Several numerical tests illustrate the effectiveness of the proposed methods. These tests show that CDA style nudging adapted to an iteration (instead of a time stepping scheme) enables convergence at much higher Rayleigh number.