Robust Arrow-Hurwicz Method for High-Rayleigh Number Boussinesq Flow

Abstract

We develop and analyze a robust Arrow-Hurwicz (AH) iterative method for the numerical solution of steady Boussinesq flows with nonhomogeneous partitioned Dirichlet boundary conditions. Although a direct AH formulation may be applied to the momentum equation alone, we demonstrate that incorporating an AH-type update for the temperature equation is crucial for stability and convergence in buoyancy-driven systems, particularly at high Rayleigh numbers. The resulting Improved Arrow-Hurwicz (IAH) scheme avoids solving saddle-point systems at each iteration and yields a fully decoupled algorithm with low computational cost per step. We establish existence, uniqueness, uniform boundedness, and convergence under standard small-data assumptions, and provide corresponding error estimates for the finite element discretization. Extensive two- and three-dimensional numerical experiments verify the theoretical findings, demonstrate significant acceleration over the alternative AH scheme and the Penalty-Picard iteration, and confirm robust convergence in high-Rayleigh number regimes. The proposed method offers a scalable and efficient solver for steady natural convection and provides a promising alternative to continuation-based approaches traditionally used for high-Rayleigh flows.