3. 7 Numerical Integration of the Source Function
75
On the other hand, a reverse transformation can be done from a finitedifference equation to a differential equation, solving it at the time step
(1- p)D.t. The equation (3.59) is rewritten in the form:
sn+l sn+p
1
-
= --(Bsn+p- Bc(so:+I)n+P).
(3.60)
(1- p)D.t
1- p
The following differential equation corresponds to it:
as= - 1 -(BS- Bcso:+I)
at 1- P
(3.61)
with the initial condition sn+p determined by (3.58). There is the analytical
solution (3.53) of (3.61) with the value B replaced by B/(1- p).
The scheme with fractional steps (3.58) and (3.59), where the second
equation is replaced by the differential equation (3.61) at p = 0, is called
a splitting or a semi-analytical method, the solution of (3.61) exists in the
analytical form (3.53). It can be used at every time step, when it is necessary
to solve the general equation (3.57).
Tests of the numerical scheme of the solution to the energy balance
equation with non-linear energy transfer function. Unlike the previous tests, it is impossible to obtain an accurate analytical solution for the
wave energy balance equation when it includes a non-linear energy transfer
function (see Sect. 4.1). That is why the most accurate numerical solution
should be used instead of the analytical one. The splitting method can be
tested by comparing the numerical solutions with the results obtained with
the help of the predictor-corrector method (3.41).
The function of the wind wave energy input is prescribed by ( 4.20), as
used in the WAM model (Komen et al., 1994). The wind speed is taken
as 18.45 ms- 1 . The dissipation is determined according to (3.50) with the
varying parameter a. At first it is assumed to be equal to 2, corresponding
to dissipation depending cubically on the spectrum. Then the calculations
are made, with this parameter dependent on the frequency function: a =
v(wfwmaxt', where v = 1, "( = 2.
According to numerical experiments, if the integration time step is assumed to be not more than 3 minutes, stable results of integrating equation
(3.56) are obtained with the help of the predictor-corrector method (3.41).
However, a constraint has to be introduced to reduce the wave energy spectral density at the high-frequency band (at frequencies greater than or equal
to double the frequency of the spectral maximum), so that its values do not
exceed the equilibrium interval.
As shown by numerical calculation, the solution obtained by the splitting
method is stable for time steps of 1, 3 and even 6 hours. That is why there is
no longer any need to introduce any solution or source function limitations.
The results of numerical calculations for the frequency spectrum at a time
moment t = 30 hours are shown in Fig. 3.7. The calculations are performed
75
On the other hand, a reverse transformation can be done from a finitedifference equation to a differential equation, solving it at the time step
(1- p)D.t. The equation (3.59) is rewritten in the form:
sn+l sn+p
1
-
= --(Bsn+p- Bc(so:+I)n+P).
(3.60)
(1- p)D.t
1- p
The following differential equation corresponds to it:
as= - 1 -(BS- Bcso:+I)
at 1- P
(3.61)
with the initial condition sn+p determined by (3.58). There is the analytical
solution (3.53) of (3.61) with the value B replaced by B/(1- p).
The scheme with fractional steps (3.58) and (3.59), where the second
equation is replaced by the differential equation (3.61) at p = 0, is called
a splitting or a semi-analytical method, the solution of (3.61) exists in the
analytical form (3.53). It can be used at every time step, when it is necessary
to solve the general equation (3.57).
Tests of the numerical scheme of the solution to the energy balance
equation with non-linear energy transfer function. Unlike the previous tests, it is impossible to obtain an accurate analytical solution for the
wave energy balance equation when it includes a non-linear energy transfer
function (see Sect. 4.1). That is why the most accurate numerical solution
should be used instead of the analytical one. The splitting method can be
tested by comparing the numerical solutions with the results obtained with
the help of the predictor-corrector method (3.41).
The function of the wind wave energy input is prescribed by ( 4.20), as
used in the WAM model (Komen et al., 1994). The wind speed is taken
as 18.45 ms- 1 . The dissipation is determined according to (3.50) with the
varying parameter a. At first it is assumed to be equal to 2, corresponding
to dissipation depending cubically on the spectrum. Then the calculations
are made, with this parameter dependent on the frequency function: a =
v(wfwmaxt', where v = 1, "( = 2.
According to numerical experiments, if the integration time step is assumed to be not more than 3 minutes, stable results of integrating equation
(3.56) are obtained with the help of the predictor-corrector method (3.41).
However, a constraint has to be introduced to reduce the wave energy spectral density at the high-frequency band (at frequencies greater than or equal
to double the frequency of the spectral maximum), so that its values do not
exceed the equilibrium interval.
As shown by numerical calculation, the solution obtained by the splitting
method is stable for time steps of 1, 3 and even 6 hours. That is why there is
no longer any need to introduce any solution or source function limitations.
The results of numerical calculations for the frequency spectrum at a time
moment t = 30 hours are shown in Fig. 3.7. The calculations are performed