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

Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations

We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics

A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

Efficient linear solvers for incompressible flow simulations using Scott--Vogelius finite elements

Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, 2012) divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. To judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and H -LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications