332
Coastal Engineering: Theory and Practice
—-----iKa$ — 0
dn
where, 9 is the wave angle. The other boundaries may represent a breakwater (an obstruction), an open boundary, or a coastline. In such cases the
following condition is imposed [Tsay and Liu, 1983].
(10.9)
where n is the direction normal to the boundary and a is a relaxation
coefficient that varies with the type of boundary and may hâve to be determined empirically. The values of “q” varies from 0 to 1 (a = 0 for full
reflection and a = 1 for full absorption). The incident wave angle, 0 in
the offshore open boundary will be obtained as the transformation from
offshore wave climate to shallow water wave angles by applying the Snell’s
law. The domain is discretized into grids of size (Az) and (A$/). A typical
finite différence grid is shown in Fig. 10.9. {$(a;i3/)} is used to donate the
grid point value of the potential, standard discretization of the Helmholtz
équation using second order finite différence scheme yields,
At2
Aî/2
2
2
- Â-ï + v'â--*' $W) = °
(10.10)
Aar
Aî/2
J k
The conventional approach consists of writing such équations for ail
points in the domain. The resulting System of équations may be expressed
in matrix form as:
[A]{$} = {/}
(10.11)
where [A] is the System matrix, {$} is the unknown vector (of the desired
grid point value of the wave potential), and {/} is a vector that contains
information about the discretized boundary condition. The direct methods like Gauss élimination requires large amount of memory to store the
System matrix values which makes it practically impossible, whereas the
itérative methods do not require the storage of matrix [A] and hence can
be used for large domains. In order to accomplish this, the matrix [A]
must be strictly diagonally dominant, or it must be symmetric and positive defmite. Conventional itérative methods like Jacobi’s or Gauss-Siedel
methods, do not guarantee convergence when applied to Eq. (10.11) since
the matrix obtained is neither diagonally dominant nor positive definitive
(because of complex quantifies in the boundary conditions). An example for
the itérative scheme is the Generalized Conjugate Gradient method, which
Coastal Engineering: Theory and Practice
—-----iKa$ — 0
dn
where, 9 is the wave angle. The other boundaries may represent a breakwater (an obstruction), an open boundary, or a coastline. In such cases the
following condition is imposed [Tsay and Liu, 1983].
(10.9)
where n is the direction normal to the boundary and a is a relaxation
coefficient that varies with the type of boundary and may hâve to be determined empirically. The values of “q” varies from 0 to 1 (a = 0 for full
reflection and a = 1 for full absorption). The incident wave angle, 0 in
the offshore open boundary will be obtained as the transformation from
offshore wave climate to shallow water wave angles by applying the Snell’s
law. The domain is discretized into grids of size (Az) and (A$/). A typical
finite différence grid is shown in Fig. 10.9. {$(a;i3/)} is used to donate the
grid point value of the potential, standard discretization of the Helmholtz
équation using second order finite différence scheme yields,
At2
Aî/2
2
2
- Â-ï + v'â--*' $W) = °
(10.10)
Aar
Aî/2
J k
The conventional approach consists of writing such équations for ail
points in the domain. The resulting System of équations may be expressed
in matrix form as:
[A]{$} = {/}
(10.11)
where [A] is the System matrix, {$} is the unknown vector (of the desired
grid point value of the wave potential), and {/} is a vector that contains
information about the discretized boundary condition. The direct methods like Gauss élimination requires large amount of memory to store the
System matrix values which makes it practically impossible, whereas the
itérative methods do not require the storage of matrix [A] and hence can
be used for large domains. In order to accomplish this, the matrix [A]
must be strictly diagonally dominant, or it must be symmetric and positive defmite. Conventional itérative methods like Jacobi’s or Gauss-Siedel
methods, do not guarantee convergence when applied to Eq. (10.11) since
the matrix obtained is neither diagonally dominant nor positive definitive
(because of complex quantifies in the boundary conditions). An example for
the itérative scheme is the Generalized Conjugate Gradient method, which