Using Equation 72 to eliminate η s yields
$
η
η
2
$
$
= - pη
$- q 2+ g
(74)
θ
n
where
η
η0
2
$
$
g = 0 + pη0 + q
$
(75)
θ2
r
Equation 74 is the open boundary condition in terms of total wave potential η .
$
For this parabolic boundary problem, the desired Jacobian functional may be more
complicated than that of Equation 19. Therefore the Galerkin finite-element formulation is
used.
According to the Galerkin approach, the wave potential η is approximated by
$
n
η = ∑ ηi N i
(76)
$
$
i=1
$
$
node i. The unknown η i can be determined from the orthogonality conditions between
$
function Ni and left-hand side of Equation 18, that is
(
)
~
∇ (~∇ η)+ b η N i dΩ = 0
(i
= 1, ..., N )
∫
∫
(77)
a $
$
Ω
Using the divergence theorem, Equation 77 becomes
28
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