Computational models are routinely applied to simulate flow and transport processes in rivers, estuaries, lakes and oceans.  Simulations are achieved by solving discrete sets of algebraic equations for the variables of interest, e.g. water surface elevation, flow rates, temperature, salinity, etc. Collectively, the solutions to these discrete sets of equations serve as an approximate solution to the relevant mathematical model, or more specifically the relevant initial-boundary value problem (IBVP), that represents the physical process(es) of interest.  An IBVP consists of the governing differential equations (e.g. the conservation equations of mass, momentum, energy, etc) posed over some limited physical domain (e.g. a lake or a section of a river) along with a suitable set of initial and boundary conditions.  Broadly speaking, there are two main approaches, or numerical methods, by which a set of discrete algebraic equations is obtained from a given continuous IBVP:  finite difference and finite element methods.

Finite difference methods, in their most basic and most commonly used form, are formulated on structured rectangular "grids".  That is, the physical domain of interest is approximated by a set of rectangles, which are most commonly equal-sized squares.  At the corners, or nodes, of each of these squares, an algebraic equation is formulated, and subsequently solved, for the variable of interest.  Briefly, these algebraic equations are formulated by replacing the continuous differential operators of the governing differential equations by "finite differences" expressed in terms of the nodal unknowns and the grid spacing Δx. One of the most attractive features of finite difference methods is the ease with which they can be implemented (and, to a certain extent, understood from a theoretical point of view) by scientists and engineers.  The main disadvantages of finite difference methods are their inability to easily handle complex geometries that one might encounter, for example, along a shoreline (see Figure 1a), and their inability to easily provide varying grid spacings within a single domain.

Finite element methods, on the other hand, can make use of what are commonly called unstructured grids.  In this case, the physical domain of interest is approximated by a set of "finite elements" comprised of triangles, quadrilaterals, or a combination of the two, of varying sizes (tetrahedra and triangular prisms are commonly used element shapes in 3D).  This allows finite element methods to handle complex geometries introduced by both the bathymetry/topography and the shoreline with relative ease (see Figure 1b).  Furthermore, unstructured grids provide a natural and computationally efficient way of resolving the large range of scales that is typically associated with flow and transport problems.  In the case of finite element methods, an approximate solution to an equivalent variational or weak form of the IBVP is constructed over each element using simple analytic functions -- most commonly linear polynomials.  This results in a discrete set of algebraic equations that can then be solved for the unknowns.  In contrast to finite difference methods, finite elements methods are initially harder to implement in a computer code due to the level of technical knowledge they require.

Historically, finite difference methods have been more extensively used in the field of computational fluid mechanics than finite element methods (interestingly, the opposite trend is observed in the field of structural mechanics).  In particular, within the field of coastal ocean and estuarine modeling, the Princeton Ocean Model (POM) -- a finite difference model -- has been extensively used for simulating circulation patterns in estuaries, coastal regions, lakes and global oceans.  POM is the hydrodynamic model used by the Great Lakes Operational Forecast System (GLOFS) developed by The Ohio State University (OSU) and the National Oceanic and Atmospheric Administration's (NOAA's) Great Lakes Environmental Research Laboratory (GLERL).  GLOFS is used operationally to provide predictions of water levels, currents and temperatures in the five Great Lakes.  It uses a 5 km by 5 km grid resolution to approximate the physical domain of the Great Lakes.

Unstructured grid models, such as those based on finite element methods, have also been used for hydrodynamic modeling, though to a lesser extent than structured models; however, due to the inherent advantages they possess over finite difference based models, their use is quickly becoming more prevalent in the area of computational fluid mechanics.  For example, the advanced circulation (ADCIRC) finite element model, originally developed at the University of Notre Dame and the University of North Carolina's Institute of Marine Sciences, is extensively used to model hurricane storm surge in the Gulf of Mexico and in particular the area of Southern Louisiana.  For these simulations, a large unstructured triangular grid is used to explicitly model the entire Western North Atlantic, the Caribbean Sea and the Gulf of Mexico -- see Figure 2.  Grid resolution varies from approximately 50 km in the deep ocean to less than 100 m in regions of Southern Louisiana.  The wide range of element sizes used in this grid demonstrates the significant advantage of using unstructured (finite element) grids over structured (finite difference) grids:  resolution can be provided where it is needed based on local flow scales and geometric complexity in order to accurately represent the problem at a minimum computational cost.

The goal of this work to develop an unstructured grid of Lake Erie that can be used with unstructured grid hydrodynamic models.  The proposed grid will provide grid resolution on the order of 1 to 5 km out in the interior of the lake down to less than 100 m in nearshore regions.  Coastal regions, such as Sandusky Bay, for example, will be explicitly discretized as part of the grid domain.  The existence of such a grid will greatly enhance the modeling capabilities in Lake Erie by offering far greater model resolution than currently available.  The grid will be particularly useful to agencies such as GLERL and the Ohio Department of Natural Resources (ODNR) for doing regional-scale studies with unstructured grid flow and transport models; examples include:  (i)  contaminate transport and water quality studies, (ii) dredging feasibility and material disposal studies, (iii) modeling wind driven circulation, and (iv) analysis of storm surge and flooding.

An unstructured finite element grid of Lake Erie has been constructed.  The grid consists of approximately 1.5 million triangular elements ranging in size from 1 km in the interior of the lake down to around 50 m in coastal areas.  The grid incorporates high-resolution shore line and bathymetric data.  This level of detailed resolution will ultimately permit more accurate simulation of Lake dynamics, especially in the nearshore. 

The following presentation will be made at the 10th US National Congress on Computational Mechanics.