## 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+i = jvi + 2hF(x„,y„) (10-28)

which requires values at x„_i and x„ to calculate _y„+i- Once we have an approximate value for y„+i, we use the corrector equation

to get an improved value of y„+i. The corrector equation is used iteratively: the value of y„+i is used to obtain an improved value of y„+i 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 xq 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 103 s~\ Again, we use a time increment of 20 seconds.

 A B C D E F 1 Simulating First-Order Reaction Using Two-point Predictor-Corrector Method 2 (Values in bold are initial values) 3 t Pred Corrl Corr2 Corr3 Corr4 4 0 0.2000 Difference between successive values (row 11) 5 20 0.1810 -3E-05 2 £-06 -3E-08 4E-09 6 40 0.16381 0.16373 0,16373 0 16373 0,16373 i 7 50 0.14821 0,14821 0.14321 0.14821 0.14821 I 8 B0 0.13417 0,13409 0,13409 0.13409 0.13409 9 100 0.12137 0.12139 0.12139 0.12139 0.12139 10 120 0.10989 0.10981 0.10981 0.10981 0.10981 11 140 0.09940 0.09943 0.09942 0.09942 0.09942 12 160 0 09001 0.08992 0.08993 0,08993 0.08993 I 13 180 0.08139 0.08144 0,08144 0,08144 0.08144 14 200 0.07373 0,07364 0.07364 0,07364 ; 0.07364

Figure 10-13. Decreasing error in the Euler method by a simple predictor-corrector method, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet 'Predictor-Corrector Method')

Figure 10-13. Decreasing error in the Euler method by a simple predictor-corrector method, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet 'Predictor-Corrector Method')

The predictor formula was entered in column B. The first two values, shown in bold, are the starting values; the predictor formula, in cell B6, corresponds exactly to equation 10-28 and is

The corrector formula, in cell C6, corresponds exactly to equation 10-29 and =\$B5+DX*(-k*\$B5-k*B6)/2

The preceding formula is used iteratively. The formula (note the use of relative and mixed references) was Filled Right to perform the iterations. The formulas in row 5 were added to display the difference between a corrected value and the preceding one (for example, the formula in cell C5 is

=B11-C11

and shows how the corrector formula converges).

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