Fast iterative methods for solving the incompressible Navier-Stokes equations
Carlos Echeverria Serur (COSSE student, double degree with TU Berlin)

Supervisor: Kees Vuik

Site of the project: TU Delft

start of the project: December 2012

In March 2013 the Interim Thesis and a presentation has been given.

The Master project has been finished in August 2013 by the completion of the Masters Thesis and a final presentation has been given.

For working address etc. we refer to our alumnipage.

Summary of the master project:
The numerical solution of the incompressible Navier-Stokes (N-S) equations is an area of much importance in contemporary scientific research. Except for some simple cases, the analytical solution of the (N-S) equations is impossible. Therefore, in order to solve these equations, it is necessary to apply numerical techniques. The most commonly used numerical discretization techniques include Finite Dierence Methods (FDM), Finite Volume Methods (FVM) and Finite Element Methods (FEM). The discretization approach followed throughout this thesis is done by the FEM. Due to the nonlinear character in the behavior of fluids, the solution of the (N-S) system of equations requires a suitable linearization of the algebraic system resulting from the FEM discretization of the original system. The resulting linear system of equations gives rise to a so-called saddle-point problem, an algebraic system which is nonsymmetric, indefinite, and typically ill conditioned.

The efficient solution of this type of linear algebraic problem is a challenge. The primary interest in these types of problems is due to the fact that most of the computing time and memory of a computational implementation is consumed by the solution of this system of equations. In this project we adopt an iterative approach to solving this linear system, mainly by the use of a Krylov subspace method combined with a preconditioned linear system of equations. In the case of the Navier-Stokes problem, the type of preconditioners studied belong to a branch of Block Preconditioners known as SIMPLE-type preconditioners (Semi Implicit Pressure Linked Equations) in literature. These methods decouple the system and solve separate subsystems of the velocity and pressure. The pressure subsystem arises from an appropriate approximation of the Schur complement of the system.

In this work, we are interested in approaching the following questions arising from the study of the saddle-point problem arising from the discretization of the Navier-Stokes equations:
• Why is there a stagnation phase in the iterative solution of the SIMPLE-preconditioned Navier-Stokes algebraic system?
• Why does the number of iterations increase for stretched grids? Wake of a ship Incompressible multi-phase flow ## Contact information: Kees Vuik Back to the home page or the Master students page of Kees Vuik