Geoscience Reference
In-Depth Information
The two equations in (4.229) can be solved separately by a direct method. After
is calculated, the new values of
can be obtained conveniently by
(
1
)
=
(
0
)
+
(4.230)
The above five-point SIP method has been extended to solve the algebraic equations
related to seven-point (Leister and Peric, 1994) and nine-point numerical schemes
(Schneider and Zedan, 1981). Because the coefficient matrices of the algebraic
equations resulting frommany numerical schemes are or can be approximated as five-,
seven-, or nine-diagonal matrices, the SIP method is often used in computational river
dynamics.
It should be noted that because the approximation used in the SIP method is related
to the discretization of partial differential equations, the SIP method makes little sense
for generic algebraic equations.
4.5.5 Over-relaxation and under-relaxation
(
1
)
can also be obtained
After the correction vector
is calculated, the new values
by the relaxation method:
(
1
)
=
(
0
)
+
α
φ
(4.231)
where
α
φ
is the relaxation factor.
α
φ
>
1 for over-relaxation, and
α
φ
<
1 for under-
relaxation.
The over- and under-relaxation methods can accelerate or decelerate the conver-
gence speed. Over-relaxation is often used in conjunction with the Gauss-Seidel
method, yielding the Successive Over-Relaxation (SOR) method. Under-relaxation
is very useful for nonlinear problems and can avoid divergence.
Faster convergence can be achieved when using an optimum value of
α
φ
. However,
because the optimum
α
φ
value depends on many factors, such as the nature of the
problem, number of grid points, grid spacing, and iteration procedure, there are no
general rules to determine it. Usually, a suitable value of
α
φ
can be found by experience
and from exploratory computation for the problem under consideration.
