5.3 Iterative Methods
107
equation is of the above type and the variants of the method just described
are often used t o solve it. AD1 methods are very commonly used when solving
compressible flow problems. They are also well adapted to parallel computation.
The method described in this section takes advantage of the structure of
the matrix which is, in turn, due to the use of a structured grid. However,
closer inspection of the development shows that the basis of the method is
an additive decomposition of the matrix:
where H is the matrix representing the terms contributed by the second
derivative with respect to x and V, the terms coming from the second derivative in the y-direction.
There is no reason why other additive decompositions cannot be used.
One useful suggestion is to consider the additive LU decomposition:
This is different from the multiplicative LU decomposition of Sect. 5.2.2.
With this decomposition, Eqs. (5.49) and (5.50) are replaced by:
Each of these steps is essentially a Gauss-Seidel iteration. The rate of convergence of this method is similar that of the AD1 method given above. It
also has the very important advantage that it does not rely on the structure
of the grid or of the matrix and may therefore be applied to problems on unstructured grids as well as structured ones. However, it does not parallelize
as well as the HV version of ADI.
5.3.6 Conjugate Gradient Methods
In this section, we present a class of methods based on techniques for solving non-linear equations. Non-linear solvers can be grouped into two broad
categories: Newton-like methods and global methods. The former converge
very quickly if an accurate estimate of the solution is available but may fail
catastrophically if the initial guess is far from the exact solution. 'Far' is a
relative term; it is different for each equation. One cannot determine whether
an estimate is 'close enough' except by trial and error. Global methods are
guaranteed to find the solution (if one exists) but are not very fast. Combinations of the two types of methods are often used; global methods are used
initially and followed by Newton-like methods as convergence is approached.
Many global methods are descent methods. These methods begin by converting the original system of equations into a minimization problem. Suppose
that the set of equations to be solved is given by Eq. (5.1) and that the matrix
107
equation is of the above type and the variants of the method just described
are often used t o solve it. AD1 methods are very commonly used when solving
compressible flow problems. They are also well adapted to parallel computation.
The method described in this section takes advantage of the structure of
the matrix which is, in turn, due to the use of a structured grid. However,
closer inspection of the development shows that the basis of the method is
an additive decomposition of the matrix:
where H is the matrix representing the terms contributed by the second
derivative with respect to x and V, the terms coming from the second derivative in the y-direction.
There is no reason why other additive decompositions cannot be used.
One useful suggestion is to consider the additive LU decomposition:
This is different from the multiplicative LU decomposition of Sect. 5.2.2.
With this decomposition, Eqs. (5.49) and (5.50) are replaced by:
Each of these steps is essentially a Gauss-Seidel iteration. The rate of convergence of this method is similar that of the AD1 method given above. It
also has the very important advantage that it does not rely on the structure
of the grid or of the matrix and may therefore be applied to problems on unstructured grids as well as structured ones. However, it does not parallelize
as well as the HV version of ADI.
5.3.6 Conjugate Gradient Methods
In this section, we present a class of methods based on techniques for solving non-linear equations. Non-linear solvers can be grouped into two broad
categories: Newton-like methods and global methods. The former converge
very quickly if an accurate estimate of the solution is available but may fail
catastrophically if the initial guess is far from the exact solution. 'Far' is a
relative term; it is different for each equation. One cannot determine whether
an estimate is 'close enough' except by trial and error. Global methods are
guaranteed to find the solution (if one exists) but are not very fast. Combinations of the two types of methods are often used; global methods are used
initially and followed by Newton-like methods as convergence is approached.
Many global methods are descent methods. These methods begin by converting the original system of equations into a minimization problem. Suppose
that the set of equations to be solved is given by Eq. (5.1) and that the matrix