Accelerating convergence of a natural convection solver by continuous data assimilation

TL;DR

The paper addresses accelerating the convergence of Picard iterations for the Boussinesq natural convection system by integrating continuous data assimilation (CDA) nudging. It proves that, with data collected at spacing $H$, the linear convergence rate is enhanced by a factor proportional to $H^{1/2}$ when nudging the velocity and temperature (or velocity alone), and it remains effective in noisy data scenarios with accuracy limited by data quality. The authors provide rigorous analyses for accurate and noisy data, derive residual and error bounds, and validate the approach through 2D and 3D numerical experiments on heated cavity benchmarks, showing convergence at higher Rayleigh numbers than Picard alone. A Newton fallback strategy is proposed to overcome noise-induced accuracy limits, highlighting practical applicability for robust simulations of natural convection.

Abstract

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 only linear and 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 data into the iteration using a continuous data assimilation (CDA) approach. We prove that our approach scales the linear convergence rate by $H^{1/2}$, where $H$ is the characteristic spacing of the measurement locations. This implies that when Picard is converging, CDA will accelerate convergence, and when Picard is not converging, CDA (with enough data) will enable convergence. In the case of noisy data, we prove that the linear convergence rate of the nonlinear residual is similarly scaled by $H^{1/2}$ but the accuracy is limited by the accuracy of the data. Several numerical tests illustrate the effectiveness of the proposed method, including when the data is noisy. These tests show that CDA style nudging adapted to an iteration (instead of a time stepping scheme) enables convergence at much higher $Ra$.

Figures (8)

• Figure 1: Shown above are the computed Boussinesq solution of the differentially heated cavity problem \ref{['num:2d']} for velocity streamlines (left) and temperature contours (right) for $Ra=1000000$
• Figure 2: Shown above are the computed Boussinesq solution of the differentially heated cavity problem\ref{['num:3d']} for velocity streamlines (left) and temperature contours (right) for $Ra=100000$
• Figure 3: Shown above are the convergence plots for the $2d$ heated cavity problem for $Ra=10000$(left) and $Ra=100000$(right) with varying $H$ for $\mu_1=\mu_2=1000.$
• Figure 4: Shown above are the convergence plots for the $3d$ heated cavity problem for $Ra=10000$ (left) and $Ra=100000$ (right) with varying $H$ for $\mu_1=\mu_2=1000.$
• Figure 5: Shown above are the convergence plots for $Ra=10000$ (left) and $Ra=100000$ (right) with varying $H$ for $\mu_1=1000, \mu_2=0$

Theorems & Definitions (36)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Remark
• Lemma 3.1
• proof
• Theorem 3.1
• Remark