Using Python to Solve the Navier-Stokes Equations - Applications in the Preconditioned Iterative Methods

This article describes a new numerical solver for the Navier-Stokes equations. The proposed solver is written in Python which is a newly developed language. The Python packages are built to solve the Navier-Stokes equations with existing libraries. We have created discretized coefficient matrices from systems of the Navier-Stokes equations by the finite difference method. In addition we focus on the preconditioned Krylov subspace iterative methods in the linearized systems. Numerical results of performances for the Preconditioned iterative methods are demonstrated. The comparison between Python and Matlab is discussed at the end of the paper

Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids

An efficient solver for the incompressible Navier–Stokes equations in rotation form

We consider preconditioned iterative methods applied to discretizations of the linearized Navier–Stokes equations in two- and three-dimensional bounded domains. Both unsteady and steady flows are considered. The equations are linearized by Picard iteration. We make use of the rotation form of the momentum equations, which has several advantages from the linear algebra point of view. We focus on a preconditioning technique based on the Hermitian/skew-Hermitian splitting of the resulting nonsymmetric saddle point matrix. We show that this technique can be implemented efficiently when the rotation form is used. We study the performance of the solvers as a function of mesh size, Reynolds number, time step, and algorithm parameters. Our results indicate that fast convergence independent of problem parameters is achieved in many cases. The preconditioner appears to be especially attractive in the case of low viscosity and for unsteady problems