54
3 Numerical Implementation of the Wave Energy Balance Equation
directions, respectively, relative to the reference grid point j. The index k
runs over the three propagation directions (3, r.p, e and the values uk, !:l.xk
denote the velocity components (3, r.p, e, (3.2)-(3.4) and the grid spacing in
the relevant directions, respectively.
The first-order scheme is characterized by a higher numerical dispersion,
with the effective diffusion coefficient D,...., !:l.x 2 I f:l.t (The WAM model, 1988).
For numerical stability the time step should satisfy the CLF condition f:l.t <
!:l.x I u, so D > f:l.xu. There is a smaller inherent numerical dispersion in the
advection term of the second-order scheme. Its disadvantage is an unphysical
negative energy generation in sharp gradient regions. This drawback can be
eliminated by including explicit diffusion terms, as indicated in (3.10). The
explicit diffusion necessary for removal of the negative side lobes in (3.10)
should be of the same order as the implicit numerical diffusion in (3.9) in
order to make the effective dispersion comparable for both schemes.
Further, the numerical results of the above scheme are compared with the
analytical and interpolation-ray method solutions of the problem.
3.4 Elimination of the "Garden Sprinkler Effect"
Kinetic equation corrections determined due to frequency-angular discretization.
In order to obtain a numerical solution of the wave
energy balance equation (3.1), a representation of the continuous spectrum
in discrete frequency-angular form is required. Let the value sn(Wk, f3z) be
a spectral component at time step n corresponding to frequency Wk and
angle (3z.
The averaged energy S(wk, f3z) in the range
{ ( wk - 0.5 !:l.w :::; w :::; wk + 0.5 !:l.w) , (f3z - 0.5 6.(3 :::; (3 :::; f3z + 0.5 6.(3)}
can be determined by integrating the continuous spectrum within this range:
(3.11a)
The usual way to estimate the integral (3.11a) is based on the rectangle
or trapezium method (Ryvkin, 1990; Ocean Wave Modeling, 1985). However, a more accurate algorithm can be applied, assuming that the value S
is a continuous and twice differentiable function. In order to estimate the
integral (3.11a), bi-quadratic interpolation including the function values at
the grid points Wk-l,wk,Wk+l and f3z-l,f3z,f3zH can be used. The cubature
formula for energy assessment is obtained by "multiplying" the quadrature
formulas of each variable (Krylov & Shulgina, 1966):
3 Numerical Implementation of the Wave Energy Balance Equation
directions, respectively, relative to the reference grid point j. The index k
runs over the three propagation directions (3, r.p, e and the values uk, !:l.xk
denote the velocity components (3, r.p, e, (3.2)-(3.4) and the grid spacing in
the relevant directions, respectively.
The first-order scheme is characterized by a higher numerical dispersion,
with the effective diffusion coefficient D,...., !:l.x 2 I f:l.t (The WAM model, 1988).
For numerical stability the time step should satisfy the CLF condition f:l.t <
!:l.x I u, so D > f:l.xu. There is a smaller inherent numerical dispersion in the
advection term of the second-order scheme. Its disadvantage is an unphysical
negative energy generation in sharp gradient regions. This drawback can be
eliminated by including explicit diffusion terms, as indicated in (3.10). The
explicit diffusion necessary for removal of the negative side lobes in (3.10)
should be of the same order as the implicit numerical diffusion in (3.9) in
order to make the effective dispersion comparable for both schemes.
Further, the numerical results of the above scheme are compared with the
analytical and interpolation-ray method solutions of the problem.
3.4 Elimination of the "Garden Sprinkler Effect"
Kinetic equation corrections determined due to frequency-angular discretization.
In order to obtain a numerical solution of the wave
energy balance equation (3.1), a representation of the continuous spectrum
in discrete frequency-angular form is required. Let the value sn(Wk, f3z) be
a spectral component at time step n corresponding to frequency Wk and
angle (3z.
The averaged energy S(wk, f3z) in the range
{ ( wk - 0.5 !:l.w :::; w :::; wk + 0.5 !:l.w) , (f3z - 0.5 6.(3 :::; (3 :::; f3z + 0.5 6.(3)}
can be determined by integrating the continuous spectrum within this range:
(3.11a)
The usual way to estimate the integral (3.11a) is based on the rectangle
or trapezium method (Ryvkin, 1990; Ocean Wave Modeling, 1985). However, a more accurate algorithm can be applied, assuming that the value S
is a continuous and twice differentiable function. In order to estimate the
integral (3.11a), bi-quadratic interpolation including the function values at
the grid points Wk-l,wk,Wk+l and f3z-l,f3z,f3zH can be used. The cubature
formula for energy assessment is obtained by "multiplying" the quadrature
formulas of each variable (Krylov & Shulgina, 1966):