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

TL;DR

This paper introduces a nonlinear preconditioning method combining Anderson acceleration with Picard iteration to enhance the stability and convergence of Newton's method in non-isothermal flow simulations, enabling higher Rayleigh number operation.

Contribution

The paper develops a novel Anderson-Picard based nonlinear preconditioning technique that improves Newton iteration stability and convergence in complex flow simulations.

Findings

Unconditional stability of Newton with Anderson-Picard preconditioning.

Quadratic convergence retained with less restrictive conditions.

Successful application to high Rayleigh number problems.

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.