A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

1. Introduction 13
The finite-difference scheme used by Casulli and Cheng (1992) is based on two time levels and uses an Eulerian-Lagrangian
method (ELM) with linear interpolation for the treatment of the advective terms in the momentum equations. The scheme uses no
coordinate transformations and solves the governing momentum equations in a nonconservative form that is consistent with the
use of an ELM. The scheme is basically an extension of an earlier 2-D model by Casulli (1990) that has been used successfully
(Cheng and others, 1993; Signell and Butman, 1992). The 3-D scheme has been used successfully for San Francisco Bay (Cheng
and Casulli, 1996). A drawback of the Casulli and Cheng (1992) scheme is that it is only first-order accurate in the time discreti-
zation, which can cause artificial damping of tidal amplitudes and velocities. Casulli and Cattani (1994) present a modified version
of the scheme in which the time discretization for the gradient of the free-surface elevation in the momentum equations and the
velocity in the continuity equation can be centered in time, according to Crank and Nicolson (1947), to achieve second-order accu-
racy for these terms. This modification does improve the overall accuracy of the scheme, but for fully nonlinear problems, the
finite-difference momentum equations are still first-order accurate in the advection and Coriolis terms and the bottom friction con-
dition. The depth-integrated continuity equation also is first-order accurate in the representation used for the total depth of flow,
which is evaluated backward-in-time. All terms in the two-level scheme could conceivably be centered in time if an iterative pro-
cedure was implemented, as will be discussed later. The scheme from Casulli and Cheng (1992) also has been implemented by
Stansby and Lloyd (1995).
In this discussion, relatively little has been said regarding 3-D models that use the finite-element method (Zienkiewicz, 1977)
for the numerical discretization in the horizontal plane. In general, the research on 3-D modeling has concentrated far more on the
finite-difference method than the finite-element method. 7 This emphasis on the finite-difference method may in part be due to a
perception that, at least until recently, finite-element models were not fully competitive with finite-difference models in terms of
computational efficiency. Because research on the finite-element method for solving free-surface flow problems started much later
than research on the finite-difference method, there has been a period in which finite-element models have been “catching up” with
the finite-difference models in terms of both accuracy and efficiency. Of course, the advantage of the finite-element method has
always been its flexibility in allowing unstructured grid networks that can be employed to represent shoreline curvature better and
to increase grid-point density where needed. This advantage has become somewhat less significant, however, with the recent
advances in curvilinear-coordinate, finite-difference models.
The progress on 3-D finite-element models followed most of the same trends as for 3-D finite-difference models. Two early
3-D models (Kawahara and others, 1982; Shubinski and Walton, 1982) use explicit time discretizations with simple elements, even
though the finite-element method does not lead naturally to explicit procedures; both these models use a multilevel, finite-differ-
ence approach in the vertical dimension. The model by Koutitas and O’Connor (1982) uses a time-splitting method (fractional step
approach) to separate the horizontal computations from the vertical computations, which are both done using finite elements. The
model by King (1985) uses the rather costly approach (in terms of computer time) of solving the nonlinear, 3-D equations using a
fully implicit, time-stepping scheme without any form of splitting procedure; to improve computational efficiency of simulations,
the model allows the convenient coupling of 1-, 2-, and 3-D elements so that the user can limit the 3-D analysis to the region in
7 Of seventeen 3-D models surveyed in a review paper by Cheng and Smith (1990), only three used the finite-element method for the numerical
discretization in the horizontal plane; the other models used the finite-difference method.