70
3 Numerical Implementation of the Wave Energy Balance Equation
where
It should be noted that a similar algorithm can be developed for the case
ofthe non-linear interaction in the source function (see Sect. 4.1). However, in
this case some additional difficulties can appear. Not only should the spectral
density for the considered wave component { wi, {3 1 } be taken into account,
but also the analogous values for other components { wi+k, f3J+n}. That is why
it is necessary to solve the corresponding non-linear set of algebraic equations
rather than one equation (3.51).
Test results of different numerical schemes.
Some of the abovementioned numerical schemes have been tested and compared with the numerical and analytical solutions. These can be obtained in the constant wind
case for the source function (3.50). The precise analytical solution is written
as follows:
S(t) = So(cS(j ± 11- cS()I e-aBt)-1/a '
(3.53)
where the ( +) sign is used for the case where the value 1 - cS<> is greater
than zero, and the (-) sign is for the opposite case.
The energy spectral density values calculated with the help of different
numerical algorithms and the analytical solution are presented in Fig. 3.4,
18 minutes after starting the evolution. The 15 ms- 1 wind speed value and
the Miles wind input formula for the parameter B are used. In the numerical
integration a 3-minute time step is implemented.
As seen in Fig. 3.4, there are essential differences between the results, in
spite of a sufficiently small time step for numerical integration. A comparison
of the numerical results with the analytical solution shows that the least
accurate are the values calculated with the help of the Euler explicit method.
It should be noted that although the results obtained according to the Adams
explicit method are in satisfactory agreement with the analytical solution,
there is an anomalous negative value at the frequency 4 rad s- 1 . The values
calculated by the semi-implicit method (3.47) and the implicit numerical
scheme (3.52) are mostly close to the analytical solution.
Further calculations were made with increased numerical integration
steps. It should be noted that the results obtained with a help of the Euler
and Adams explicit schemes are unstable, especially in the high-frequency
band. That is why they are no longer used.
The results using an analytical and three numerical schemes with a 20minute integration step at the time moment of 3 hours are presented in
Fig. 3.5. The initial stage of the calculations is characterized by the overestimated spectral density value calculated using the implicit scheme (3.52). The
underestimated values are calculated by the semi-implicit method (3.47) and
3 Numerical Implementation of the Wave Energy Balance Equation
where
It should be noted that a similar algorithm can be developed for the case
ofthe non-linear interaction in the source function (see Sect. 4.1). However, in
this case some additional difficulties can appear. Not only should the spectral
density for the considered wave component { wi, {3 1 } be taken into account,
but also the analogous values for other components { wi+k, f3J+n}. That is why
it is necessary to solve the corresponding non-linear set of algebraic equations
rather than one equation (3.51).
Test results of different numerical schemes.
Some of the abovementioned numerical schemes have been tested and compared with the numerical and analytical solutions. These can be obtained in the constant wind
case for the source function (3.50). The precise analytical solution is written
as follows:
S(t) = So(cS(j ± 11- cS()I e-aBt)-1/a '
(3.53)
where the ( +) sign is used for the case where the value 1 - cS<> is greater
than zero, and the (-) sign is for the opposite case.
The energy spectral density values calculated with the help of different
numerical algorithms and the analytical solution are presented in Fig. 3.4,
18 minutes after starting the evolution. The 15 ms- 1 wind speed value and
the Miles wind input formula for the parameter B are used. In the numerical
integration a 3-minute time step is implemented.
As seen in Fig. 3.4, there are essential differences between the results, in
spite of a sufficiently small time step for numerical integration. A comparison
of the numerical results with the analytical solution shows that the least
accurate are the values calculated with the help of the Euler explicit method.
It should be noted that although the results obtained according to the Adams
explicit method are in satisfactory agreement with the analytical solution,
there is an anomalous negative value at the frequency 4 rad s- 1 . The values
calculated by the semi-implicit method (3.47) and the implicit numerical
scheme (3.52) are mostly close to the analytical solution.
Further calculations were made with increased numerical integration
steps. It should be noted that the results obtained with a help of the Euler
and Adams explicit schemes are unstable, especially in the high-frequency
band. That is why they are no longer used.
The results using an analytical and three numerical schemes with a 20minute integration step at the time moment of 3 hours are presented in
Fig. 3.5. The initial stage of the calculations is characterized by the overestimated spectral density value calculated using the implicit scheme (3.52). The
underestimated values are calculated by the semi-implicit method (3.47) and