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arXiv · 2501.08855

A simple-to-implement nonlinear preconditioning of Newton's method for solving the steady Navier-Stokes equations

Muhammad Mohebujjaman·Mengying Xiao·Cheng Zhang
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Abstract

The Newton's method for solving stationary Navier-Stokes equations (NSE) is known to convergent fast, however, may fail due to a bad initial guess. This work presents a simple-to-implement nonlinear preconditioning of Newton's iteration, that remains the quadratic convergence and enlarges the domain of convergence. The proposed AAPicard-Newton method adds the Anderson accelerated Picard step at each iteration of Newton's method for solving NSE, which has been shown globally stable for the relaxation parameter $\beta_{k+1}\equiv1$ in the Anderson acceleration optimization step, convergent quadratically, and converges faster with a smaller convergence rate for large Reynolds number. Several benchmark numerical tests have been tested and are well-aligned with the theoretical results.

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Muhammad Mohebujjaman, Mengying Xiao, Cheng Zhang. 2025-01-15. A simple-to-implement nonlinear preconditioning of Newton's method for solving the steady Navier-Stokes equations. https://arxiv.org/abs/2501.08855

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