74
3 Numerical Implementation of the Wave Energy Balance Equation
trum. Additional solution limitations are introduced in its numerical implementation (The WAM model, 1988; Komen et al., 1994). As noted, such
limitations are not physically well-grounded. The application of the implicit
scheme is difficult for numerical integration of the wave energy balance equation due to the complicated expression of the non-linear interaction. In this
case either the explicit schemes (3.40)-(3.41) or the semi-implicit scheme
(3.47) can be used.
In this section an attempt is undertaken to consider a more efficient alternative scheme than those mentioned above. For this purpose the splitting
method is introduced.
The total source function G is supposed to include the wind wave energy
input, non-linear dissipation and non-linear energy transfer Gnl· It can be
written as:
G(S) = BS(1 - cS 0 ) + Gn, .
(3.55)
In order to solve the problem of the evolution of the spectrum in time,
the energy balance equation is expressed in the form:
~~ = G(S).
(3.56)
The numerical form of (3.56) is as follows:
(3.57)
This scheme can be used for any value p: 0 :<:::: p :<:::: 1. It should be remembered
that there is the Euler explicit scheme for p = 0, an implicit scheme for p = 1
and it is assumed that p = 1/2 for the rectangle method.
Now the scheme (3.57) with fractional steps is considered. At the first step
the variation in the solution due to non-linear interactions can be calculated
as:
sn+p- sn
--6.-t-- = Gnl ·
(3.58)
At the second step the variation in the solution due to the remaining part
of the source function can be determined:
snH sn+p
;;.t = Bsn+p - Bc(sa+l t+P .
(3.59)
If sn+p is deduced from the first equation (3.58) and substituted into the
left-hand part of the second equation (3.59) or the sum of these two equations
by terms, the scheme with fractional steps is reduced to the scheme (3.57).
The finite difference scheme (3.59) is actually a simple Euler one, requiring
a small time step compared to other known schemes.
3 Numerical Implementation of the Wave Energy Balance Equation
trum. Additional solution limitations are introduced in its numerical implementation (The WAM model, 1988; Komen et al., 1994). As noted, such
limitations are not physically well-grounded. The application of the implicit
scheme is difficult for numerical integration of the wave energy balance equation due to the complicated expression of the non-linear interaction. In this
case either the explicit schemes (3.40)-(3.41) or the semi-implicit scheme
(3.47) can be used.
In this section an attempt is undertaken to consider a more efficient alternative scheme than those mentioned above. For this purpose the splitting
method is introduced.
The total source function G is supposed to include the wind wave energy
input, non-linear dissipation and non-linear energy transfer Gnl· It can be
written as:
G(S) = BS(1 - cS 0 ) + Gn, .
(3.55)
In order to solve the problem of the evolution of the spectrum in time,
the energy balance equation is expressed in the form:
~~ = G(S).
(3.56)
The numerical form of (3.56) is as follows:
(3.57)
This scheme can be used for any value p: 0 :<:::: p :<:::: 1. It should be remembered
that there is the Euler explicit scheme for p = 0, an implicit scheme for p = 1
and it is assumed that p = 1/2 for the rectangle method.
Now the scheme (3.57) with fractional steps is considered. At the first step
the variation in the solution due to non-linear interactions can be calculated
as:
sn+p- sn
--6.-t-- = Gnl ·
(3.58)
At the second step the variation in the solution due to the remaining part
of the source function can be determined:
snH sn+p
;;.t = Bsn+p - Bc(sa+l t+P .
(3.59)
If sn+p is deduced from the first equation (3.58) and substituted into the
left-hand part of the second equation (3.59) or the sum of these two equations
by terms, the scheme with fractional steps is reduced to the scheme (3.57).
The finite difference scheme (3.59) is actually a simple Euler one, requiring
a small time step compared to other known schemes.