Internal generation of waves for extended Boussinesq equations

158
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C. Lee et al.rCoastal Engineering 42 2001 155–162
rector scheme. Both cnoidal and sinusoidal waves
are generated internally from two viewpoints of the
mass transport and energy transport.
3.1. Finite difference method
Sponge layers are placed at boundaries in order to
minimize wave reflections from the boundaries. Eqs.
Ž. 1 and Ž. 2 can be written in a horizontally one-dimensional
domain as,
½ž /
Eh E E z 2 a h 2 E 2 u
q Ž hqh. u4
q y h
2
Et Ex Ex 2 6 Ex
ž /
h E 2
q z q h Ž hu. s0 Ž 10.
2 Ex
a 2 5
½ ž / 5
Eu 1 Eu 2 Eh za
E 3 u E 2 Eu
q qg qza q h
2 2
Et 2 Ex Ex 2 Ex Et Ex Et
qv Dus0, s
Ž 11.
where,
° d r S
~ e y1 , inside sponge layer
Ds s ey1
. Ž 12.
¢ 0, outside sponge layer
In Eq. Ž 12 ., Ss2.5 L Ž L is the wavelength.
is the
thickness of the sponge layer and d is the distance
from the starting point of the sponge layer.
The variables h and u in Eqs. Ž 10. and Ž 11.
iy1r2 i
are placed in a staggered grid system where the
subscript i denotes the spatial grid point. The firstorder
spatial derivative terms in Eqs. Ž 10. and Ž 11.
are discretized to OŽŽ D x . 4 .. Both the second-and
third-order spatial derivative terms in Eq. Ž 10.
are
discretized to OŽŽ D x. 2 . and the second-order spatial
derivative terms in Eq. Ž 11.
are discretized to
OŽŽ D x . 4 ..
The time derivative terms in Eqs. Ž 10. and Ž 11.
are discretized to OŽŽ Dt. 2 . using the predictor-corrector
method. Eq. Ž 10.
is discretized explicitly between
the n-th and Ž nq1r2.
-th time steps to get the
p
surface elevation h at the Ž nq1r2.
-th time step.
The superscript p in h stands for the predictor. Eq.
Ž 10.
is discretized again implicitly between the n-th
and Ž nq1r2.
-th time steps to get the surface elevanq1r2
tion h at the Ž nq1r2.
-th time step. At this
time, the surface elevation h and the particle velocity
u in the spatial derivatives are replaced by h p
and Žu n qu nq1 . r2, respectively. And then, Eq. Ž 11.
is discretized between the n-th and Ž nq1.
-th time
steps. At this time, the surface elevation h is replaced
by h nq1r2 . The resulting equation gives a
hepta-diagonal matrix of u nq1 given as:
Au nq1 qBu nq1 qCu nq1 qDu nq1 qEu nq1
i iy3 i iy2 i iy1 i i i iq1
qFiuiq2 nq1 qGu i iq3 nq1 sH i , Ž 13.
where A i, ..., Hi
are coefficients of h n and u n .
Using the method of LU decomposition ŽPress et al.,
1992 ., Eq. Ž 13.
is solved to get the particle velocity
nq1
u . Again, Eq. Ž 10.
is discretized between the n-th
and Ž nq1.
-th time steps. At this time, the particle
velocity u is replaced by Žu n qu nq1 . r2 and the
surface elevation h in EŽ hqh. u4
i
rEx is replaced
Ž
n nq1 n nq1
by h qh qh qh .
iq1r2 iq1r2 iy1r2 iy1r2 r4. The re-
sulting equation gives a penta-diagonal matrix given
as,
ah nq1 qbh nq1 qgh nq1 qdh nq1
i iy5r2 i iy3r2 i iy1r2 i iq1r2
qeihiq3r2 nq1 ss i , Ž 14.
where a i, ..., si
are coefficients of h n , u n , and
u nq1 . Using the method of LU decomposition, Eq.
Ž .
nq1
14 is solved to get the surface elevation h .
Waves are generated smoothly by multiplying
Ž . Ž .
)
tanh nDtrT T is the wave period to h in Eq.
Ž. 3 . At both the left and right boundaries, a reflective
boundary condition is specified as:
u nq1 s0, u nq1 s0, h nq1 sh nq1 ,
1 Iy1 1r2 3r2
h nq1 sh nq1 , Ž 15.
Iy1r2
Iy3r2
where is1 and isIy1 denote the points of the
left and right boundaries, respectively. The grid size
is chosen as D xsLr20 in order to get sufficient
spatial resolution. The time step is chosen so that the
Courant number is CrsCe
DtrD xs0.2 and a sta-
ble solution is guaranteed.