Page 258 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 10 ORDINARY DIFFERENTIAL EQUATIONS. PART I 235

Predictor-Corrector Methods
The methods in the preceding sections are one-step methods. They need only
the value of the preceding point to calculate the value of the new point. Thus
they are self-starting methods. Predictor-corrector methods, on the other hand,
use the values of two or more previous points to calculate the value of the new
point. They are not self-starting; two or more known initial values are needed.
Often a Runge-Kutta calculation is used to provide the needed values.
Predictor-corrector methods use two formulas, the predictor equation and the
corrector equation. There are many forms of predictor and corrector equations,
but all operate according to the same principle: calculate an approximate value of
the function using a predictor equation, then use a corrector equation to correct
the value.

A Simple Predictor-Corrector Method

To illustrate the method we will modify the simple Euler method, equation
10-6, as follows. The predictor equation is

Yn+l = Y,-l + 2hF(x,,,y,) (1 0-28)
which requires values at x,-t and x, to calculate Y,+~. Once we have an
approximate value for Y,+~, we use the corrector equation

( 10-29)

to get an improved value of yn+l. The corrector equation is used iteratively: the
value of y,+l is used to obtain an improved value of y,+l and the process is
continued until a specified level of convergence is obtained. Two starting values
are required, and generally only a single value at xo is provided as part of the
statement of the problem; the fourth-order Runge-Kutta method can be used to
obtain the other starting value.
The worksheet shown in Figure 10-13 illustrates the application of this
simple predictor-corrector formula. Again we use as an example the simulation
of the first-order kinetic process A -+ B with initial concentration CO = 0.2000
mol/L and rate constant k = 5 x s-'. Again, we use a time increment of 20
seconds.