next up previous contents
Next: Computer-based methods of constructing Up: Introduction Previous: Introduction

Modelling requirements for irregular grids

In studying the behaviour of tides, storm surges or tsunamis in coastal waters by means of numerical models, there are several advantages in being able to vary the size and orientation of elements in the grid on which the model is based. In the first place, it is possible to obtain a much better fit to the coastlines when the edges of grid elements can be made of arbitrary length and direction. In contrast, for regular grids with uniform rectangular elements of the type used for most finite difference schemes, the edges of elements are necessarily parallel to the two axes, with the result that some parts of the coastline have to be approximated somewhat roughly by 'staircase' arrangements of line segments.

Use of irregular grids also facilitates better model design in other practical and theoretical respects. Considerable computing time is wasted when a grid with uniform element size is used in situations where water depth varies substantially over the area modelled. If an explicit difference scheme is used, the Courant stability criterion requires that cT/L 1, where c = wave speed, T = time step, and L is some linear measure of element size, such as length or width. Since c increases with water depth and it is impractical to use different sizes of time step at different parts of the grid, having L uniform over the whole grid leads to unnecessarily frequent computation everywhere except at the point of maximum depth. Most implicit schemes are stable even if the Courant criterion is violated, but truncation error generally increases with the local value of cT/L ; so again, if L is uniform over the grid, some unnecessary computing is carried out over the shallower parts of the model domain.

For these reasons, it is desirable to design grids whose elements change in size according to water depth so as to keep the local Courant ratio cT/L fairly uniform over the whole grid. Since wave speed c for shallow water waves is related to local water depth by the formula c2 = gd , the above condition is equivalent to requiring that the spatial sampling interval (number of grid points per wavelength) should be kept as nearly uniform as possible over the whole grid. Many modellers assume that this is the optimum condition for element size in shallow-water models. For instance, LePrevost and Vincent [1986] use a grid with between 13 and 14 points per wavelength as the standard for comparison of different grids in a test case with large depth variation. Quite possibly, numerical experiments are the only means of establishing the optimality of uniform sampling per wavelength in irregular grids, though perhaps some form of statistical proof may be possible.

So far, triangular elements have generally been preferred over quadrilateral elements for shallow-water models. The choice depends on the numerical scheme to be used later with the grid. Since shallow-water models are generally based on the essentially first-order primitive equations, low-order basis functions appear to be adequate and, consequently, triangular elements are appropriate.

It is common practice to make the individual elements in triangular grids as nearly equilateral as possible, on the grounds that this should reduce truncation errors in the subsequent numerical solutions. On the other hand, some writers feel that this condition is not important [Simpson, 1979]. The arguments presented are necessarily dependent on the numerical scheme used. Foreman [1984] found when comparing particular schemes on regular grids composed of equilateral or isosceles triangles that the equilateral grids allowed isotropic wave propagation, whereas in grids of isosceles triangles, wave speed varied with directon. It was decided here to aim at making all triangles as nearly equilateral as possible, on the assumption that this is likely to prove desirable for some numerical schemes and seems unlikely to impair the qualities of others.

Another consideration in constructing triangular grids is the number of elements which can be permitted to meet at a grid vertex. This question is linked to that of the preferred shape of triangle. For a grid composed of regular equilateral triangles, six elements meet at each vertex. For regular grids of right-angled isosceles triangles, there can be either six elements per vertex or alternately four and eight elements. Platzman [1980] found that the inhomogenity in the latter triangulation generated numerical noise, that is, small-scale errors. Grids produced by the methods outlined in this report very rarely have fewer than five elements meeting at any interior vertex; no upper limit on the number of elements per vertex is imposed, but use of the Renka triangulation algorithm seldom results in more than nine elements per vertex.

An additional criteria is that the size of the elements should vary smoothly over the spatial region. That is, it is undesirable to have small elements adjacent to large elements as this configuration increases the truncation errors and hence reduces the accuracy of most numerical approximation schemes. In practical terms, area ratios between adjacent elements should not exceed approximately 2. In general, the grid generation method described later leads to smooth changes in element size. However, the programs include a variety of tests to verify the quality of the grid.

On the assumption that no triangle in the grid diverges very much from equilateral shape, it is possible to recast the requirement for near-uniform cT/L in a convenient form. Squaring this ratio and bearing in mind that the area of an equilateral triangle is proportional to the square of its linear dimension, the above condition translates into the requirement that c2t2/A should be near-uniform, where A is element area. Since c2 = gd, and T is normally uniform for the whole grid, this means essentially that d / A, the ratio of element area to local water depth, should be kept as uniform as possible over the grid.

next up previous contents
Next: Computer-based methods of constructing Up: Introduction Previous: Introduction

Channel Consulting Ltd.