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

1. Introduction 15
Presently, the differences in the computational efficiencies of the new finite-difference, the ADI (Leendertse, 1989; Uitten-
bogaard and others, 1992), and the semi-implicit (Casulli and Cheng, 1992) methods are difficult to assess and depend on the
geometry and hydrodynamic conditions of the particular water body under study. As de Goede (1992) points out, semi-implicit
methods require solving a pentadiagonal matrix system using a two-dimensional structure, whereas the ADI schemes solve only
simple tridiagonal systems. The ADI schemes, however, may be limited because of geometry to a certain maximum time step for
accuracy (the so-called ADI effect). The semi-implicit methods do not suffer from the ADI effect, but they may have a known
time-step limitation related to the explicit treatment of other terms. The dependence on geometry and bathymetry by the ADI
scheme is a disadvantage because errors due to the ADI effect can go undetected by an unaware user and cause misinterpretation
of model results. Ultimately the decision was made to implement the semi-implicit approach in the new model. To simulate the
hydrodynamics of a geometrically complex estuary, such as the Suisun Bay region of San Francisco Bay, the semi-implicit scheme
may indeed be the most efficient approach.
The semi-implicit scheme implemented herein closely follows the approach outlined by Casulli and Cheng (1992) for the
implicit inclusion of vertical diffusion in a 3-D calculation without recourse to mode-splitting. The details of the overall scheme,
however, differ from those in Casulli and Cheng (1992) in many significant ways. The time integration used here is a three-time-
level, semi-implicit, leapfrog-trapezoidal method. Except for a small amount of first-order error introduced by the uncentered (in
time) treatment of horizontal diffusion for stability considerations, the scheme is second-order accurate in the truncation errors of
the finite-difference approximations in both space and time. All terms (other than horizontal diffusion), as well as nonlinear coef-
ficients, are centered in time during the leapfrog and trapezoidal steps of the scheme to achieve second-order accuracy in time. The
accurate evaluation of the nonlinearities in the equation of motion is important, because nonlinearities have a significant effect on
the tidally averaged circulation in shallow estuaries. The governing equations for the multilevel scheme are prepared in a conser-
vative form by integrating them over the height of each layer. The layer-integrated, volumetric transports replace velocities as the
dependent variables so that the depth-integrated continuity equation that is used in the solution for the water surface elevation is
linear. The advection terms in the momentum and salt-transport equations are solved using explicit, leapfrog-trapezoidal integra-
tion rather than an ELM as used by Casulli and Cheng (1992). The leapfrog-trapezoidal approach does very well with the conser-
vation of salt, which can be a problem with the ELM approach. The global matrix solution for water surface elevation in the new
model is implemented using the conjugate-gradient method with a modified incomplete Cholesky preconditioner (Axelsson and
Lindskog, 1986). This method was selected after testing a number of other preconditioners and acceleration methods.