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

2025/01/15 by Mohebujjaman, Muhammad, Xiao, Mengying, Zhang, Cheng
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2501.08855

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 β_k+1≡1 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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