# Discretization

The original version of FVCOM was developed in the σ-coordinate transformation system. The code was subsequently upgraded to the generalized terrain-following coordinate system in 2006. The discretization forms of governing equations have been significantly modified in this new coordinate system. When the non-hydrostatic version of FVCOM was developed, we implemented a semi-implicit solver, so the current version of FVCOM has two options for the time integration: 1) mode-split and 2) semi-implicit. In this chapter, we provide an example of the discrete forms of the hydrostatic FVCOM in the σ-coordinate transformation system for the mode-split solver. The σ-coordinate transformation is one selection of the generalized terrain-following coordinates, so learning the details of the discretization forms in this coordinate system can make users it easy to learn how the generalized terrain-following coordinates work in FVCOM. A brief description of the semi-implicit solver is given in Chapter 4 when the non-hydrostatic solver is introduced. Users, who are interested in learning the details of discretization forms in the generalized terrain-following coordinate system, can examine the source code directly.

## Discretization Stencil

Similar to a triangular finite element method, the horizontal numerical computational domain is subdivided into a set of non-overlapping unstructured triangular cells. An unstructured triangle is comprised of three nodes, a centroid, and three sides (Fig. 3.1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be expressed as:

$[X(i),Y(i)], \; \; i=1:N$

and the location of the nodes can be specified as:

$[Xn(j),Yn(j)], \;\; j=1:M$

Since none of the triangles in the grid overlap, N should also be the total number of triangles. On each triangular cell, the three nodes are identified using integral numbers defined as $N_i(\hat{j})$ where $\hat{j}$ is counted clockwise from 1 to 3. The surounding triangles that have a common side are counted using integral numbers defined as where is counted clockwise from 1 to 3. At open or coastal solid boundaries, is specified as zero. At each node, the total number of the surrounding triangles with a connection to this node is expressed as , and they are counted using integral numbers where m is counted clockwise from 1 to .