(1) should be developed with reliable accuracy. That is why it is necessary to use sufficiently accurate and fast

numerical algorithm, not available at that time.

It should be noted that a full -scale experiment on numerical simulation of the equation (1) is performed recently

(Pushkarev et al., 2002). The approach, based on another numerical algorithm (Webb 1978; Resio&Perrie, 1991),

existing for more than two decades, is used. As this problem seems to be rather complicated for numerical

simulations and the results are obtained for the first time, there appears a necessity to produce independent

estimations in order to verify the above-mentioned results with the help of the another algorithm.

This is a motivation for writing present paper. Numerical results of the equation (1) solution taking into account

wave energy forcing and damping are obtained in this paper with the help of the numerical algorithm elaborated

recently by one of the authors (Lavrenov 1998, 2001). The algorithm is based on numerical methods of highest

precision. It is of high accuracy and calculation speed.

2. PROBLEM FORMULATION

The equation of spectral wave action evolution can be written as:

+*γ*

=G +F

(9)

where

*C ω * sin 2 ((1 - *ω ω * ) (π 2)), *if ω *< *ω*

1 min

min

min

(10),

(

)

2

*C * 2ω p ω ω p -1 , *if ω *> *ω * p

where C1,2 are positive constants.

=

The value F is an external active force:

. The function

within the frequency range:

. It is equal to the following angular function:

*Q *cos n ( *A β *), *if * cos( *A β *) > 0 and *ω*min < *ω *< *ω * f

,

(11)

0, in another cases

where *Q *is a normalizing function, providin g the same integral value for the various parameters *n *and *A:*

π

∫ f (β, ω)*d*β = *C *ω

3

(12)

-π

where C3 is a constant.

So, a problem is posed in such a way that the whole frequency range is divided into sub -ranges with

different energy sources. The low-frequency damping domain is located in the range (ω < *ω *in ) in order to

m

stabilize fluxes directed into the low frequency range. The energy pumping domain is located within the range

( ωmin < *ω *< *ω*f ). One of the most interesting frequency ranges is the domain (ω f < *ω *< *ω * p ) without any

damping or pumping, i.e. it is the so-called "transparency window", where the spectrum is formed only by non-linear