Grid point placement

Placement of a probe at each of the grid points and calculating interaction energies. 

One interesting research issue currently receiving much attention is placement of grid points for the discretization. For a single distributed variable such as temperature, one can see how to place the points, i.e., put more points where the variable is changing more rapidly and fewer where it is not. How is this to be done for many distributed but coupled variables that are changing at quite different rates in different parts of the space This problem is similar to that of slow- and fast-moving units in a dynamic simulation, only here no natural modularity occurs within a flowsheet of interconnected units. 

Figure 1.11 Placement of grid points for finite difference computation.

The remaining matter requiring attention is the placement of the breakpoints. Similar considerations will apply here as outlined in Section 4.2(a) in connection with the finite-difference grid point distribution. That is, it is important to concentrate the breakpoints at the position where most reaction occurs, with progressively lower interior breakpoint concentrations towards the flame boundaries. For (presumably) an Eulerian calculation of an ozone... 

Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal (x) and vertical (y) axis. These two axes intersect at one coordinate point—(0,0)—the origin. A coordinate pair, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and second number representing the vertical. Coordinate points are given in the form of (x,y). 

The above has considered arbitrarily spaced grids, whereas in practice, the spacing is often that of Feldberg [231]. In terms of points, the special case is a sequence of positions given by an exponentially expanding series of spatial intervals. This will be detailed in Chap. 7. Here, it is sufficient to mention that this special case makes the derivation of the coefficients for various derivative approximations easier and the expressions themselves more compact, as was reported by Martfnez-Ortiz [385]. That author also found that there is a particular value for the expansion parameter, q = /2j for which the asymmetric four-point second derivative, referred to the second of the four points u"(2,4), is third-order accurate, rather than second-order as for other parameter values or arbitrary placement of points. This can be of use in simulation. The four-point approximation has some good properties besides this, as will be explained in Chap. 7. 

As well, it will be seen that the formulation of the second spatial derivative on a general grid, spaced in some unspecified way, is rather flexible and permits easy replacement, in a given program, of the stretching function used including, if one desires, equal spacing, or even arbitrary placement of each point. 

A stretched stack of boxes was used by Seeber and Stefani and by Feldberg [7, 8] for the box-method, to be described in Chap. 9. Pao and Dougherty [17] developed the same idea (and stretching function) in 1969, in the context of fluid dynamic simulations. This is the simple placement of points at increasing intervals, in some suitable point distribution or stretching function, and discretisation of the second derivative of concentration along X on that unequal grid. 

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