The N-S equations feature non-linearity due to the presence of the advection term in the momentum conservation equation.
The decoupling between the velocity and the pressure in these equations also brings in extra complexity.
So, the N-S equations can only be solved iteratively with an initial guess of their solution, which leads to two types of methods:
The decoupled methods: using the SIMPLE-like algorithms, the equations of mass and momentum conservation are solved one by one using several sets of algebraic equations, and the algorithms achieve coupling of equations and convergence of the iterations in a prediction-correction manner. These methods seems slow (i.e., hundreds or even thousands of iterations are typically required to get accurate results).
The fully coupled methods: methods like the Newton-Raphson's method (or the inexact Newton's methods such as the Jacobian-free Krylov subspace method, JNFK) solve the N-S equations simultaneously by assembling a much larger set of nonlinear systems that output the solutions of velocity and pressure in each nonlinear iteration, and may achieve a much higher rate of convergence (local quadratic convergence, the Picard method do the same thing but typically converges more slowly).
However, to my knowledge, the Newton's method seems common when solving N-S equations with the finite element method (ref.1) or finite difference method but not so prevalent for the finite volume method (some textbooks stick to the SIMPLE-like methods, e.g., Versteeg, 2007*, ref.2)
Is there any drawback for the Newton’s method when dealing with the N-S equations with FVM?
Refs:
Versteeg H., 2007, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd Edition. Loughborough University.
In most cases, the incompressible Navier-Stokes equations cannot be solved analytically.
One way to discretize the problem is to reformulate it in a weak formulation by using the Finite Element Method. Here, a non-linear term within the incompressible NavierStokes equations poses a challenge. One can approach the non-linearity using splitting schemes, or use techniques for solving non-linear equations, such as fixed-point iteration (also called Picard iteration) or Newton’s Method. Also, stabilization methods are applied in order to make the discretized problem numerically stable.
One method involving stabilization is the cG(1)cG(1)-method introduced by Eriksson.
In Michael Brandl's thesis, he introduced an implementation of Newton’s method for the cG(1)cG(1)method of numerically solving the incompressible Navier-Stokes equations, and compared it to an earlier version using fixed-point iteration. The velocity and pressure results from both methods correspond closely, as investigated by qualitative analysis by considering streamlines and xz velocity profiles, as well as considering the kinetic energy as a quantitative measure for velocity. For the pressure, surface visualizations have been used for quantitative comparisons. Newton’s Method in this implementation does not converge quadratically as hoped for, and has a much higher computation time than the existing method using splitting and fixed-point iteration.