Solving the System of Algebraic Equations

Abstract

The result of the discretization process is a system of linear equations of the form \( {\mathbf{A}}{\varvec{\upphi}} = {\mathbf{b}} \) where the unknowns \( {\varvec{\upphi}} \), located at the centroids of the mesh elements, are the sought after values. In this system, the coefficients of the unknown variables constituting matrix A are the result of the linearization procedure and the mesh geometry, while vector b contains all sources, constants, boundary conditions, and non-linearizable components. Techniques for solving linear systems of equations are generally grouped into direct and iterative methods, with many sub-groups in each category. Since flow problems are highly non-linear, the coefficients resulting from their linearization process are generally solution dependent. For this reason and since an accurate solution is not needed at each iteration, direct methods have been rarely used in CFD applications. Iterative methods on the other hand have been more popular because they are more suited for this type of applications requiring lower computational cost per iteration and lower memory. The chapter starts by presenting few direct methods applicable to structured and/or unstructured grids (Gauss elimination, LU factorization, Tridiagonal and Pentadiagonal matrix algorithms) to set the ground for discussing the more widely used iterative methods in CFD applications. Then the performance and limitations of some of the basic iterative methods with and without preconditioning are reviewed. This include the Jacobi, Gauss-Siedel, Incomplete LU factorization, and the conjugate gradient methods. This is followed by an introduction to the multigrid method that is generally used in combination with iterative solvers to help addressing some of their important limitations.