(7)
Orthogonal Collocation for symmetric problems
For some reaction diffusion problems it is possible to show that the solution is an even function of x. For example, for the problem
(7)
it is possible to prove that the solution involves only even powers of x. In such cases it is convenient to have an orthogonal collocation method that takes this feature into account. This can easily be done by involving expansions that only involve even powers of x. Thus take the expansion
which is equivalent to
The polynomials are defined to be orthogonal with the weighting function W(x2).
(8)
where the power on xa-1 defines the geometry as planar or cartesian (a=1), cylindrical (a=2) and spherical (a=3). We follow an analogous development to obtain the (N+1) x (N+1) matrices
In addition we have the quadrature formula
where
The collocation points for a typical case (N = 3) are shown in the figure.
Collocation points for symmetric polynomials
As an example, take the problem
We apply orthogonal collocation at the interior points
and solve for the boundary condition.
The boundary conditon at x = 0 is satisfied automatically by the trial function. After obtaining the solution we obtain the effectiveness factor by calculating
Note the effectiveness factor is the average reaction rate divided by the reaction rate evaluated at the external conditions.
The collocation points are listed in Table II. For small N the results are usually more accurate when the weighting function in Eq. (8) is 1 x2. The matrices for N=1 and N=2 are given in the Table III for the three geometries. Computer programs to generate matrices are available [Finlayson, p. 325, 1980] (link). A program to solve reaction diffusion problems, OCRXN, is also available [Finlayson, p. 331, 1980].
If the solution is desired at the center (a frequent situation since the center concentration can be the most extreme one), it is given by
Error bounds have been given for linear problems [Michelsen and Villadsen, p. 356, 1981]. For planar geometry the error is
This method is very accurate for small N (and small f2); note that for finite difference methods the error goes as 1/N2, which does not decrease as rapidly with N.