N-point derivations

There are two ways at arriving at these approximations Taylor expansions and polynomials. They give the same results but the former method also gives an idea of errors. The following treatment has been given previously (Britz 1987). We start with the point-method distribution. 

For a three-point approximation, cut the polynomial off after the 2 term, successively substitute c Cq, then c c, c = C2 and solve for a. This will be found somewhat easier than the Taylor expansion method and gives the same results. The process can be automated for a computer. Resulting expressions can be written in the general form 

In Chapt. 6, these approximations will be discussed the results indicate that if we want g as accurate as the simulated concentration values, it is best to use the five-point approximation, Eq. 4.88. It may be thought that an expression like Eq. 4.88 is a lot of trouble and will increase computing time fortunately, however, in simulations, the 

the Taylor expansions become a little more unwieldy  

A final remark here is that Eq. 4.85 (or 4.86) is applicable, in that form, to both point- and box-methods (only the coefficients are different). 

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