Discretization


Computer models must represent fluid behavior with finite amount of information. This reduction from infinite to finite is discretization. Three different aspects of discretization are involved in building a numerical model:
We will deal with the first aspect of discretization later; in this lecture we will discuss methods for discretizing continuous fields. I will mention four techniques:
How we discretize the field would determine how we would construct the algorithm for the model.

Grid Points

The most straightforward way to discretize a continuous field, such as concentration of a chemical, is to evaluate it at finite number of distinct locations called grid points.

Structured and unstructured grid

Typically, these points are organized into a criss-cross pattern (grid or mesh); in two dimensions it resembles a matrix, and in three dimensions it would resemble a crystal lattice. Each grid point can then be designated with two or three indices respectively. Such a grid is called structured.
A grid can also be unstructured: it may have no discernible structure other than a degree of uniformity of spacing. Such grid points can be designated with a single ordered index, but which two points would interact does not follow a straightforward pattern.

Obviously an unstructured grid is more flexible; however, it is much simpler to formulate a numerical model over a structured grid, and a structured-grid code is also easier to optimize through a compiler (since the order in which gridded information is to be processed is more obvious).

Control Volumes

Another way to discretize a continuum is to divide the underlying space into cells called control volumes. This is a particularly intuitive way to discretize the concentration field, since we can then formulate the model in terms of the amount of the concentrate, or inventory, in each control volume, and its budget. Control volumes can also be structured or unstructured; often, the average concentration for the volume is assigned at the centroid point of the volume, giving the model a hybrid character that would also let us use the finite difference as well as finite volume method to construct the numerical algorithm.

Finite Elements

In finite element discretization, space is again divided into small cells or elements. However, within each element the field is still variable; the variability is represented in terms of a simple function, such as a linear gradient. The field will be made continuous across the boundary between elements. The effect is rather like representing a continuous surface with a number of flat facets.

Finite element discretization is particularly well-suited to unstructured grids, and is a very popular approach in these cases where the geometry is too complicated for a structured grid.

Spectral methods

The previous three methods one way or the other discretize the space itself. An altogether different approach is to construct the field as a weighted sum of known functions, such as a truncated Fourier series.

Spectral representation has some highly desirable mathematical properties; for instance, using the spectral method it is possible to arrive at algorithms that would globally minimize the discretization error, giving us a model that is optimal in accuracy. Also, when we use functions that are eigenfunctions of some of the terms in the equation (e.g. trigonometric functions for second derivatives), the algorithm can be greatly simplified (but other terms, such as non-linear terms, could become greatly complex).

The main drawback of a spectral representation is that it requires the underlying domain to be of relatively simple shape like a rectangle or a sphere (or it could be mapped easily to simple geometries). Because of this, spectral models are rarely used outside of idealized, process-oriented experimental models such as direct simulation of turbulence.

We shall not discuss spectral methods further, since its application in oceanography is rather limited.

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