72
3 Numerical Implementation of the Wave Energy Balance Equation
s
0.40
0.30
--€r -1
-+- -2
-a- -3
0.20
-+- -4
0.10
0.00
0.00
1.00
2.00
3.00
ro, rad s -t
Fig. 3.5. Calculated spectral density values by different numerical algorithms with
integration step fit = 20 minutes at 3 hours: 1 - analytical solution; 2 - semiimplicit method (3.47); 3- optimal order method (3.49); 4- implicit scheme (3.52).
Dotted line denotes the equilibrium interval values
where the sum is calculated for all spectral components: frequencies (j =
1, ... , N) and directions (k = 1, ... , M). The index i denotes the results for
different numerical schemes.
It should be noted that normalization by the maximum spectral density
of the analytical solution in (3.54) yields "an estimate from below". The
calculation error would be much greater if it were normalized by the current
spectral density value.
The changes of calculation errors in time for different methods and the
integration time step of 1 hour are shown in Fig. 3.6.
The general tendency of decreasing calculation error is related with numerical values, approaching the analytical solution and increase of the maximum spectral density, used for normalizing the calculation error in (3.54).
The relative changes in the calculation error turned out to be the same
in comparison with the previous calculations. The most accurate calculations
came from the implicit numerical scheme (3.52). With the exception of the
first integration steps, there are obvious advantages of the scheme (3.52)
compared with the other ones within the entire calculation range.
The numerical experiments reveal that the Adams scheme (3.41) is
the most accurate among the explicit schemes (3.40, 3.41). However, these
3 Numerical Implementation of the Wave Energy Balance Equation
s
0.40
0.30
--€r -1
-+- -2
-a- -3
0.20
-+- -4
0.10
0.00
0.00
1.00
2.00
3.00
ro, rad s -t
Fig. 3.5. Calculated spectral density values by different numerical algorithms with
integration step fit = 20 minutes at 3 hours: 1 - analytical solution; 2 - semiimplicit method (3.47); 3- optimal order method (3.49); 4- implicit scheme (3.52).
Dotted line denotes the equilibrium interval values
where the sum is calculated for all spectral components: frequencies (j =
1, ... , N) and directions (k = 1, ... , M). The index i denotes the results for
different numerical schemes.
It should be noted that normalization by the maximum spectral density
of the analytical solution in (3.54) yields "an estimate from below". The
calculation error would be much greater if it were normalized by the current
spectral density value.
The changes of calculation errors in time for different methods and the
integration time step of 1 hour are shown in Fig. 3.6.
The general tendency of decreasing calculation error is related with numerical values, approaching the analytical solution and increase of the maximum spectral density, used for normalizing the calculation error in (3.54).
The relative changes in the calculation error turned out to be the same
in comparison with the previous calculations. The most accurate calculations
came from the implicit numerical scheme (3.52). With the exception of the
first integration steps, there are obvious advantages of the scheme (3.52)
compared with the other ones within the entire calculation range.
The numerical experiments reveal that the Adams scheme (3.41) is
the most accurate among the explicit schemes (3.40, 3.41). However, these