5.3 Iterative Methods
97
5.3 Iterative Methods
5.3.1 Basic Concept
Any system of equations can be solved by Gauss elimination or LU decomposition. Unfortunately, the triangular factors of sparse matrices are not sparse,
so the cost of these methods is quite high. Furthermore, the discretization
error is usually much larger than the accuracy of the computer arithmetic so
there is no reason to solve the system that accurately. Solution to somewhat
more accuracy than that of the discretization scheme suffices.
This leaves an opening for iterative methods. They are used out of necessity for non-linear problems, but they are just as valuable for sparse linear
systems. In an iterative method, one guesses a solution, and uses the equation to systematically improve it. If each iteration is cheap and the number
of iterations is small, an iterative solver may cost less than a direct method.
In CFD problems this is usually the case.
Consider the matrix problem represented by Eq. (5.1) which might result
from FD or FV approximation of a flow problem. After n iterations we have
an approximate solution 4n which does not satisfy these equations exactly.
Instead, there is a non-zero residual pn:
By subtracting this equation from Eq. (5.1), we obtain a relation between
the iteration error defined by:
where + is the converged solution, and the residual:
The purpose of the iteration procedure is to drive the residual to zero; in
the process, E also becomes zero. To see how this can be done, consider an
iterative scheme for a linear system; such a scheme can be written:
M + ~ + '
= N@ln + B .
(5.16)
An obvious property that must be demanded of an iterative method is that
the converged result satisfies Eq. (5.1). Since, by definition, at convergence,
+n+l = 4' ' = 4, we must have:
A = M - N
and B = Q ,
(5.17)
or, more generally,
P A = M - N
and B = P Q ,
(5.18)
where P is a non-singular pre-conditioning matrix.
An alternative version of this iterative method may be obtained by subtracting M @ from each side of Eq. (5.16). We obtain:
97
5.3 Iterative Methods
5.3.1 Basic Concept
Any system of equations can be solved by Gauss elimination or LU decomposition. Unfortunately, the triangular factors of sparse matrices are not sparse,
so the cost of these methods is quite high. Furthermore, the discretization
error is usually much larger than the accuracy of the computer arithmetic so
there is no reason to solve the system that accurately. Solution to somewhat
more accuracy than that of the discretization scheme suffices.
This leaves an opening for iterative methods. They are used out of necessity for non-linear problems, but they are just as valuable for sparse linear
systems. In an iterative method, one guesses a solution, and uses the equation to systematically improve it. If each iteration is cheap and the number
of iterations is small, an iterative solver may cost less than a direct method.
In CFD problems this is usually the case.
Consider the matrix problem represented by Eq. (5.1) which might result
from FD or FV approximation of a flow problem. After n iterations we have
an approximate solution 4n which does not satisfy these equations exactly.
Instead, there is a non-zero residual pn:
By subtracting this equation from Eq. (5.1), we obtain a relation between
the iteration error defined by:
where + is the converged solution, and the residual:
The purpose of the iteration procedure is to drive the residual to zero; in
the process, E also becomes zero. To see how this can be done, consider an
iterative scheme for a linear system; such a scheme can be written:
M + ~ + '
= N@ln + B .
(5.16)
An obvious property that must be demanded of an iterative method is that
the converged result satisfies Eq. (5.1). Since, by definition, at convergence,
+n+l = 4' ' = 4, we must have:
A = M - N
and B = Q ,
(5.17)
or, more generally,
P A = M - N
and B = P Q ,
(5.18)
where P is a non-singular pre-conditioning matrix.
An alternative version of this iterative method may be obtained by subtracting M @ from each side of Eq. (5.16). We obtain: