78
3 Numerical Implementation of the Wave Energy Balance Equation
the wind field cannot be considered to be uniform and stationary for waves
travelling distances in one large time step.
The use of the interpolation-ray method as well as the effective numerical
method of source function integration facilitates the solution of the problem
and essentially enlarges the numerical integration time step. Obviously, the
time step can be limited by the solution accuracy as well as by the time scale
of the wind field variation.
As the first step it is possible to determine the initial wave coordinates
('!9?, cp~), which arrived at the regular grid point( '!9;, 'Pj ), using (3.2)-(3.4).
These initial coordinates ( ' !9?, cp~) may not coincide with a regular grid point.
The polynomial interpolation (3.27) is used to determine the initial value of
the spectrum S?j at the point ( ' !9?, cp~). In case the source function G is not
equal to zero, the spectral density is not constant along the trajectory of the
wave packet propagation. In order to integrate numerically the source function along the characteristic, one of the above described predictor-corrector
(3.41), semi-implicit (3.47) or splitting (3.58), (3.59) methods can be used.
It is important to take into account that at the initial moment of every
time step at the point ( ' !9?, cp~) the energy spectral density S~, the source
function G?j and the wind speed U are determined by interpolation (3.27)
at t = tn. But at a finite step moment these values are taken in the point
('!9;, 'Pj) at t = tn+l·
In conclusion it should be noted that this method of solution can also be
easily generalized for the general case (1.84), (1.86)-(1.90), i.e. in a spherical
surface with non-uniform currents and a basin with an uneven bottom. For
practical calculations, the sets of coordinates and the angles of harmonic propagation should be previously calculated for every regular grid point, taking
into account wave refraction in the current and shallow water. Their typical
scale variation should be much larger than the spacing between the numerical
grid points. At the next step the numerical calculation of the wave energy
balance equation can be made more quickly, as the solution is reduced to
interpolation with the coefficients calculated beforehand.
3.8 Conclusions
The implementation of an optimal numerical scheme for the wave energy balance equation is considered in this chapter. A sufficiently effective algorithm
for the solution of the problem has been elaborated using the fact that at every time step the solution can be divided into calculations of the wave energy
propagation and the spectral density change due to the total source function. An attempt has been made to apply the accurate analytical solution in
combination with the interpolation methods and the simplest finite-difference
schemes.
The examples of numerical calculations of swell propagation reveal that
the first-order numerical scheme of solving the wave energy balance equation
3 Numerical Implementation of the Wave Energy Balance Equation
the wind field cannot be considered to be uniform and stationary for waves
travelling distances in one large time step.
The use of the interpolation-ray method as well as the effective numerical
method of source function integration facilitates the solution of the problem
and essentially enlarges the numerical integration time step. Obviously, the
time step can be limited by the solution accuracy as well as by the time scale
of the wind field variation.
As the first step it is possible to determine the initial wave coordinates
('!9?, cp~), which arrived at the regular grid point( '!9;, 'Pj ), using (3.2)-(3.4).
These initial coordinates ( ' !9?, cp~) may not coincide with a regular grid point.
The polynomial interpolation (3.27) is used to determine the initial value of
the spectrum S?j at the point ( ' !9?, cp~). In case the source function G is not
equal to zero, the spectral density is not constant along the trajectory of the
wave packet propagation. In order to integrate numerically the source function along the characteristic, one of the above described predictor-corrector
(3.41), semi-implicit (3.47) or splitting (3.58), (3.59) methods can be used.
It is important to take into account that at the initial moment of every
time step at the point ( ' !9?, cp~) the energy spectral density S~, the source
function G?j and the wind speed U are determined by interpolation (3.27)
at t = tn. But at a finite step moment these values are taken in the point
('!9;, 'Pj) at t = tn+l·
In conclusion it should be noted that this method of solution can also be
easily generalized for the general case (1.84), (1.86)-(1.90), i.e. in a spherical
surface with non-uniform currents and a basin with an uneven bottom. For
practical calculations, the sets of coordinates and the angles of harmonic propagation should be previously calculated for every regular grid point, taking
into account wave refraction in the current and shallow water. Their typical
scale variation should be much larger than the spacing between the numerical
grid points. At the next step the numerical calculation of the wave energy
balance equation can be made more quickly, as the solution is reduced to
interpolation with the coefficients calculated beforehand.
3.8 Conclusions
The implementation of an optimal numerical scheme for the wave energy balance equation is considered in this chapter. A sufficiently effective algorithm
for the solution of the problem has been elaborated using the fact that at every time step the solution can be divided into calculations of the wave energy
propagation and the spectral density change due to the total source function. An attempt has been made to apply the accurate analytical solution in
combination with the interpolation methods and the simplest finite-difference
schemes.
The examples of numerical calculations of swell propagation reveal that
the first-order numerical scheme of solving the wave energy balance equation