Page 243 - Excel for Scientists and Engineers: Numerical Methods

220 EXCEL NUMERICAL METHODS

10-1 with the analytical expression for the concentration, [A], = in
column C. At the end of approximately one half-life (seven cycles of calculation
in this example), the error has already increased to 3.6%. Accuracy can be
increased by decreasing the size of At, but only at the expense of increased
computation. A much more efficient way of increasing the accuracy is by means
of a series expansion. The Runge-Kutta methods, which are described next,
comprise the most commonly used approach.

The Fourth-Order Runge-Kutta Method
The Runge-Kutta methods for numerical solution of the differential equation
dyldx = F(x, y) involve, in effect, the evaluation of the differential function at
intermediate points between x, and x,,+~. The value of ynCl is obtained by
appropriate summation of the intermediate terms in a single equation. The most
widely used Runge-Kutta formula involves terms evaluated at x,,, x,+~/x/~ and
x,,+~. The fourth-order Runge-Kutta equations for dyldx = F(x, y) are

Ti + 2Tz + 2T3 + T4
Yn+l =Yn + Ax (1 0-7)
6
where

(1 0-9)

Ax
T3 = F(x,+ --,yn+ -) (1 0-1 0)
T2
2 2
T4 = F (xn + b, ~n + T3) (10-1 1)
If more than one variable appears in the expression, then each is corrected by
using its own set of TI to T4 terms.

Fourth-Order Runge-Kutta Method
Implemented on a Worksheet

The spreadsheet in Figure 10-2 illustrates the use of the RK method to
simulate the first-order kinetic process A + By again using initial concentration
[Ale = 0.2000 and rate constant k = 5 x The differential equation is, again,
equation 10-4. This equation is of the simple form dyldx = F(y), and thus only
they, terms of TI to T4 need to be evaluated. The FK terms (note that TI is the
Euler method term) are shown in equations 10-12 through 10-15.