64
3 Numerical Implementation of the Wave Energy Balance Equation
in some points the error comprises 90 per cent at t = 48 hours. In the lateral
directions, the underestimation of numerical values comprises 15-20 per cent
at t = 12 hours to 40 per cent at t = 48 hours of wave propagation.
Increasing the number of directions to 24 reduces the error level approximately by a factor of two (see Fig. 3.2b) in the numerical scheme (3.9). The
results of the INTERPOL method with 12 directions (see Fig. 3.2c) are in
better agreement with the analytical solution compared to the results of the
scheme (3.9). Its accuracy is comparable to that of the scheme (3.9) with
24 directions. The INTERPOL method with 24 directions further reduces
the error level by a factor of two.
As mentioned above the time step is not limited by the CLF criterion
in the INTERPOL method. Actually, its accuracy could be improved by
applying larger time steps. This can be shown by comparing the error levels
of the INTERPOL method for time steps of 20 minutes and 6 hours (see
Fig. 3.2c,d).
The parameters of the integral solution (the temporal evolution of the
total wave energy, position of the wave field centre, degree of energy diffusion
and root mean square error (3.32)-(3.34), (3.37)) are presented in Fig. 3.3.
As shown, both numerical schemes are able to reproduce the value of the
solution of the first two mentioned parameters (see Fig. 3.3a,b). The results
of the INTERPOL method show a slightly larger dispersion than the results
of the scheme (3.9). The energy distribution obtained in the grid area by the
INTERPOL method is closer to the exact solution, as can be seen from the
results of the energy diffusion function 8(t) (see Fig. 3.3c).
It is interesting to consider the root mean square (RMS) error (see
Fig. 3.3d). There is a monotonic error increase for all numerical methods
in the initial wave propagation stages (up to 24 hours). After that the error begins decreasing due to the wave area extending beyond the numerical
grid. The RMS error (for the entire numerical area) obtained by the scheme
(3.9) with 12 directions comprises 8 per cent in 12 hours and 20 per cent in
40 hours. The shift of the basic directions by !:1(3 /2 leads to approximately
the same error. A two-fold increase in the number of directions decreases the
level of errors approximately by half for the middle stage of wave propagation. The INTERPOL method with 12 directions and the same 20 minute
time step results in a 5 per cent error at t = 12 hours and 12.5 per cent at
t = 40 hours. However, the increase of the time step to 3 hours gives a 3 per
cent error at t = 12 hours and 11 per cent error at t = 40 hours. The 6-hour
time step gives a 10 per cent error at t = 40 hours. Thus, the error level
decreases with increasing time step.
In order to study the dependence of the model results on the spatial form
of the initial perturbation, the calculations for a number of functions and
the extent of their decrease from the initial perturbation centre are repeated.
It turns out that in the prescribed numerical grid the form of the initial
perturbation does not significantly influence the wave height distribution and
evolution. Qualitatively, the characteristic features of the spatial wave height
3 Numerical Implementation of the Wave Energy Balance Equation
in some points the error comprises 90 per cent at t = 48 hours. In the lateral
directions, the underestimation of numerical values comprises 15-20 per cent
at t = 12 hours to 40 per cent at t = 48 hours of wave propagation.
Increasing the number of directions to 24 reduces the error level approximately by a factor of two (see Fig. 3.2b) in the numerical scheme (3.9). The
results of the INTERPOL method with 12 directions (see Fig. 3.2c) are in
better agreement with the analytical solution compared to the results of the
scheme (3.9). Its accuracy is comparable to that of the scheme (3.9) with
24 directions. The INTERPOL method with 24 directions further reduces
the error level by a factor of two.
As mentioned above the time step is not limited by the CLF criterion
in the INTERPOL method. Actually, its accuracy could be improved by
applying larger time steps. This can be shown by comparing the error levels
of the INTERPOL method for time steps of 20 minutes and 6 hours (see
Fig. 3.2c,d).
The parameters of the integral solution (the temporal evolution of the
total wave energy, position of the wave field centre, degree of energy diffusion
and root mean square error (3.32)-(3.34), (3.37)) are presented in Fig. 3.3.
As shown, both numerical schemes are able to reproduce the value of the
solution of the first two mentioned parameters (see Fig. 3.3a,b). The results
of the INTERPOL method show a slightly larger dispersion than the results
of the scheme (3.9). The energy distribution obtained in the grid area by the
INTERPOL method is closer to the exact solution, as can be seen from the
results of the energy diffusion function 8(t) (see Fig. 3.3c).
It is interesting to consider the root mean square (RMS) error (see
Fig. 3.3d). There is a monotonic error increase for all numerical methods
in the initial wave propagation stages (up to 24 hours). After that the error begins decreasing due to the wave area extending beyond the numerical
grid. The RMS error (for the entire numerical area) obtained by the scheme
(3.9) with 12 directions comprises 8 per cent in 12 hours and 20 per cent in
40 hours. The shift of the basic directions by !:1(3 /2 leads to approximately
the same error. A two-fold increase in the number of directions decreases the
level of errors approximately by half for the middle stage of wave propagation. The INTERPOL method with 12 directions and the same 20 minute
time step results in a 5 per cent error at t = 12 hours and 12.5 per cent at
t = 40 hours. However, the increase of the time step to 3 hours gives a 3 per
cent error at t = 12 hours and 11 per cent error at t = 40 hours. The 6-hour
time step gives a 10 per cent error at t = 40 hours. Thus, the error level
decreases with increasing time step.
In order to study the dependence of the model results on the spatial form
of the initial perturbation, the calculations for a number of functions and
the extent of their decrease from the initial perturbation centre are repeated.
It turns out that in the prescribed numerical grid the form of the initial
perturbation does not significantly influence the wave height distribution and
evolution. Qualitatively, the characteristic features of the spatial wave height