1.
Analytical Solution 
The geometric
structure of the channel is shown in Fig. 1, where is the water depth that decreases linearly toward the end
of the channel, is
the water depth of the open boundary of the channel, and are the
distances from the origin to the end and mouth of the channel,
and B is the width of the channel. 

Fig
1. The illustration of the semienclosed channel
for the tidal wave test case. O. B.: Open boundary; B. W.:
Boundary wall at the end of the channel; B is the width
of the channel; and Ho is the mean depth at the open boundary 


(3.1) 


Rewriting that and (3.1) yields, 

(3.2) 

Specifying a periodic tidal forcing with amplitude of A at
the mouth of the channel, i.e., 

(3.3) 

and a noflux boundary condition at the end of the channel,
i.e., 

(3.4) 

the analytical solution of (3.2) is given as 

(3.5) 

where 

(3.6) 

J_{0 }and Y_{0} are zero
and firstorder Bessel functions, and . 
2.
Design of the Numerical Experiment 
The nature of the oscillation described in (3.5)
depends on the geometry of the semienclosed channel. Two cases
are tested here:
1. Normal (nonresonance) case: B = 5km, H_{0}=20 m, L = 290 km, and H(L_{1})
= 10 m;
2. Nearresonance case: B = 5km, H_{0}=20
m, L = 290 km, and H(L_{1}) = 0.67 m
Both cases are driven by the M2 tidal forcing [frequency:
ω = 2Π / (12.42 X 3600 sec); amplitude: =
1 cm] at the open boundary 

Fig
2. Unstructured triangular and structured rectangular
grids used for FVCOM and POM/ECOMsi. The horizontal resolution
is 2.5 km. 

Numerical experiments are designed for a 2D case,
in which no advections, crosschannel variation, and mixing are
taken into account. Numerical grids of FVCOM and POM/ECOMsi are
constructed by triangular and square meshes with a horizontal resolution
of 2.5 km, respectively. Since the crosschannel current is zero
everywhere, only two triangular and square grid cells in the crosschannel
section are needed to calculate the water elevation and alongchannel
transport for either FVCOM or POM/ECOMsi. To avoid artificial oscillations
caused by an impulse of tidal forcing at open boundary, the elevation
and current at each grid at the initial are specified according
to the analytical solution 
3.
Results 
In the normal
case, given small tidal amplitude of 1 cm at the mouth of
the channel, is characterized with a node point at the middle of the channel
and a maximum value of 1.2 cm at the end of the channel. No
resonance can happen in this case.
The amplitude and phase of the M2 tidal wave
computed by FVCOM, POM, and ECOMsi are identical, all of
them are in accurate agreement with the analytical solution
(Fig. 3).
The analytical solution describes a standing
wave with a node point at the center of the channel. This
feature is accurately reproduced by all the three models no
matter what numerical schemes are used. 

Fig
3. Comparison of the modelpredicted and analytical
amplitudes and phases in the alongchannel direction under
a nonresonance geometric condition for FVCOM, POM and ECOMsi.
The solid line is the analytical solution and dashed line
is the model simulation. The origin of the coordinate is located
at the end of the channel. In this case, the initial distributions
of currents and sea surface are specified using the analytical
solution. 

In the nearresonance case, however, the elevation
computed by POM grows exponentially with time (Fig. 4). Numerical
instability eventually causes the model to blow up after 30 tidal
cycles, even though the basic pattern of the alongchannel distribution
of the elevation remained unchanged. This instability can not be
suppressed by reducing the time step. The semiimplicit implicit
scheme used in ECOMsi ensures numerical stability, this method,
however, causes a considerable numerical oscillation relative to
the exact solution and reverses the phase after 20 tidal cycles.
Unlike POM and ECOMsi, FVCOM remains numerically stable at all
times. The amplitude of the elevation computed by FVCOM accurately
matches the exact solution, although the phase at the node point
shows a small oscillation relative to the exact value.
Theoretically speaking, for
these particular rectangular channel cases, the finitevolume
and finitedifference approaches should be identical, since
they all are designed to conserve the mass in individual volumes.
The difference in the performances of POM, ECOMsi, and FVCOM
for the nearresonance case is believed to be due to particular
numerical methods used in these models.
The standard version of POM uses a central
difference scheme for both time integration and spatial gradients
of surface elevation and volume transport. Although it remains
a second order accuracy for numerical computation, it requires
time and space smoothing to avoid numerical instability. 

Fig
4. Comparison of the modelpredicted and analytical
amplitudes and phases in the alongchannel direction under
a nearresonance geometric condition for FVCOM, POM and ECOMsi.
The solid line is the analytical solution and dashed lines
are the model results. TC: Tidal cycle. In this case, the
initial distributions of currents and sea surface are specified
using the analytical solution. 

Since this model depends heavily on an artificial
smoothing procedure to suppress the growth of unbalanced masses,
it is always questioned whether or not this approach is suitable
for a longterm integration regarding an issue of mass conservation.
ECOMsi uses a semiimplicit scheme to release the restriction of
time step used to calculate the surface gravity waves. This method,
however, could lead to a numerical decay of tidal energy for improper
selection of time step. It is clear that the numerical method used
in ECOMsi fails to simulate the tidal wave under nearresonance
condition in the shallow water, even though it could always keep
numerical computations stable. FVCOM is solved numerically using
an integrated form of the momentum equations with an approach of
flux estimation at secondorder accuracy (Chen et al., 2003). This
method seems better than methods used for POM and ECOMsi for an
unusual case with large tidal oscillations in shallow water. 