Anderson-Picard based nonlinear preconditioning of the Newton iteration for non-isothermal flow simulations

TL;DR

The paper proposes and analyzes an Anderson-accelerated Picard nonlinear preconditioner for Newton's method applied to the non-isothermal (Boussinesq) flow system. By applying an Anderson-accelerated Picard step before the Newton iteration, the method achieves unconditional stability under a data-uniqueness condition and retains quadratic convergence with significantly relaxed small-data and initial-guess requirements, while incurring only modest additional cost due to decoupling the heat equation. Theoretical analysis shows that the Picard step improves Newton's stability and enlarges the convergence basin, and numerical experiments on a differentially heated cavity and a complex domain validate these claims, demonstrating substantially higher permissible Rayleigh numbers $Ra$ than either Picard or Newton alone. The results indicate that nonlinear preconditioning via AA-Picard-Newton yields a robust, efficient solver for high-$Ra$ non-isothermal flows and can be extended to AA variants.

Abstract

We propose, analyze, and test a nonlinear preconditioning technique to improve the Newton iteration for non-isothermal flow simulations. We prove that by first applying an Anderson accelerated Picard step, Newton becomes unconditionally stable (under a uniqueness condition on the data) and its quadratic convergence is retained but has less restrictive sufficient conditions on the Rayleigh number and initial condition's accuracy. Since the Anderson-Picard step decouples the equations in the system, this nonlinear preconditioning adds relatively little extra cost to the Newton iteration (which does not decouple the equations). Our numerical tests illustrate this quadratic convergence and stability on multiple benchmark problems. Furthermore, the tests show convergence for significantly higher Rayleigh number than both Picard and Newton, which illustrates the larger convergence basin of Anderson-Picard based nonlinear preconditioned Newton that the theory predicts.

Figures (8)

• Figure 1: Shown above is the computed solution of the differentially heated cavity problem for velocity streamlines (left) and temperature contours (right) for $Ra=10000$
• Figure 2: Shown above are convergence plots for Picard-Newton at varying $Ra$.
• Figure 3: Shown above are convergence plots for Picard (left) and Newton(right) at varying $Ra$.
• Figure 4: Convergence plots for AA-Picard-Newton with $m=1$ (left) and $m=3$ (right) at varying $Ra$.
• Figure 5: Shown above is the computed solution for $u$ (left) and $T$ (right) for $Ra=1000$ in the second numerical test.

Theorems & Definitions (23)

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