5. Solution of Linear Equation Systems
5.1 Introduction
In the previous two chapters we showed how the convection-diffusion equation
may be discretized using FD and FV methods. In either case, the result of
the discretization process is a system of algebraic equations, which are linear
or non-linear according to the nature of the partial differential equation(s)
from which they are derived. In the non-linear case, the discretized equations
must be solved by an iterative technique that involves guessing a solution,
linearizing the equations about that solution, and improving the solution;
the process is repeated until a converged result is obtained. So, whether the
equations are linear or not, efficient methods for solving linear systems of
algebraic equations are needed.
The matrices derived from partial differential equations are always sparse,
i.e. most of their elements are zero. Some methods for solving the equations
that arise when structured grids are used will be described below; all of the
non-zero elements of the matrices then lie on a small number of well-defined
diagonals; we may take advantage of this structure. Some of the methods are
applicable to matrices arising from unstructured grids as well.
The structure of the coefficient matrix for a 2D problem discretized with
a five point approximation (upwind or central difference) is shown in Fig. 3.5.
The algebraic equation for one CV or grid node is given by Eq. (3.42), and
the matrix version of the complete problem is given by Eq. (3.43), see Sect.
3.8, which is repeated here:
In addition to describing some of the better solution methods for linear
algebraic systems representing discretized partial differential equations, we
shall discuss the solution of non-linear systems of equations in this chapter.
However, we begin with linear equations. It is assumed that the reader has
had some contact with methods for solving linear systems so the descriptions
are brief.
5.1 Introduction
In the previous two chapters we showed how the convection-diffusion equation
may be discretized using FD and FV methods. In either case, the result of
the discretization process is a system of algebraic equations, which are linear
or non-linear according to the nature of the partial differential equation(s)
from which they are derived. In the non-linear case, the discretized equations
must be solved by an iterative technique that involves guessing a solution,
linearizing the equations about that solution, and improving the solution;
the process is repeated until a converged result is obtained. So, whether the
equations are linear or not, efficient methods for solving linear systems of
algebraic equations are needed.
The matrices derived from partial differential equations are always sparse,
i.e. most of their elements are zero. Some methods for solving the equations
that arise when structured grids are used will be described below; all of the
non-zero elements of the matrices then lie on a small number of well-defined
diagonals; we may take advantage of this structure. Some of the methods are
applicable to matrices arising from unstructured grids as well.
The structure of the coefficient matrix for a 2D problem discretized with
a five point approximation (upwind or central difference) is shown in Fig. 3.5.
The algebraic equation for one CV or grid node is given by Eq. (3.42), and
the matrix version of the complete problem is given by Eq. (3.43), see Sect.
3.8, which is repeated here:
In addition to describing some of the better solution methods for linear
algebraic systems representing discretized partial differential equations, we
shall discuss the solution of non-linear systems of equations in this chapter.
However, we begin with linear equations. It is assumed that the reader has
had some contact with methods for solving linear systems so the descriptions
are brief.