Question 7.8: ABM4 PREDICTOR–CORRECTOR METHOD Consider the initial-value p...
ABM4 PREDICTOR–CORRECTOR METHOD
Consider the initial-value problem in Examples 7.4 through 7.7
y′ − x²y = 2x² , y(0) = 1, 0 ≤ x ≤ 1, h = 0.1
Compute the estimated value of y_4 = y(0.4) using the ABM4 predictor–corrector method.
f(x, y) = x²(2 + y). The first element y_0 is given by the initial condition. The next three are obtained by RK4 as
y_1 = 1.001000, y_2 = 1.008011, y_3 = 1.027122The respective f(x, y) values are calculated next
f_0=f(x_0,y_0)=f(0,1)=0, f_1=f(x_1,y_1)=f(0.1,1.001000)=0.030010
f_2=f(x_2,y_2)=f(0.2,1.008011)=0.120320, f_3=f(x_3,y_3)=f(0.3,1.027122)=0.272441
Prediction
Equation 7.42 yields
y^{(1)}_{i+1}=y_i+\frac{1}{24}h(55f_i-59f_{i-1}+37f_{i-2}-9f_{i-3}), i=3,4, \cdot \cdot \cdot ,n (7.42)
y^{(1)}_{4}=y_3+\frac{1}{24}h(55f_3-59f_{2}+37f_{1}-9f_{0})=1.064604
f^{(1)}_{4}=f(x_4,y^{(1)}_{4} ) =f(0.4,1.064604)=0.490337
Correction
Equation 7.43 yields
y^{(k+1)}_{i+1}=y_i+\frac{1}{24}h(9f^{(k)}_{i+1}+19f_{i}-5f_{i-1}+9f_{i-2}),k=1,2,3,\cdot \cdot \cdot (7.43)
y^{(2)}_{4}=y_3+\frac{1}{24}h(9f^{(1)}_{4}+19f_{3}-5f_{2}+9f_{1})=1.064696, Rel error=0.0008%
This corrected value may be improved by substituting y^{(2)}_{4} and the corresponding f^{(2)}_{4}=f(x_4,y^{(2)}_{4} ) into Equation 7.43,
y^{(3)}_{4}=y_3+\frac{1}{24}h(9f^{(2)}_{4}+19f_{3}-5f_{2}+9f_{1})and inspecting the accuracy. In the present analysis, we perform only one correction so that y^{(2)}_{4}is regarded as the value that will be used for y_4. This estimate is then used in Equation 7.42 with the index i incremented from 3 to 4. Continuing this process, we generate the numerical results in Table 7.4.
| TABLE 7.4 | |||
| Summary of Calculations in Example 7.8 | |||
| x | y_{\text{RK4}} |
\overset{\backsim }{y_{i} } Predicted |
y_1 Corrected |
| 0.0 | 1.000000 | ||
| 0.1 | 1.001000 | ||
| 0.2 | 1.008011 | ||
| 0.3 | 1.027122 | Start ABM4 | |
| 0.4 | 1.064604 | 1.064696 | |
| 0.5 | 1.127517 | 1.127662 | |
| 0.6 | 1.223795 | 1.224004 | |
| 0.7 | 1.363143 | 1.363439 | |
| 0.8 | 1.557958 | 1.558381 | |
| 0.9 | 1.824733 | 1.825350 | |
| 1.0 | 2.186134 | 2.187052 | |
Another well-known predictor–corrector method is the fourth-order Milne’s method:
Predictor y^{(1)}_{i+1}=y_{i-3}+\frac{4}{3}h(2f_i-f_{i-1}+2f_{i-2}), i=3,4, \cdot \cdot \cdot ,n
Corrector y^{(k+1)}_{i+1}=y_{i-1}+\frac{1}{3}h(f^{k} _{i+1}+4f_{i}+f_{i-1}), k=1,2,3, \cdot \cdot \cdot
where f^{(k)}_{i+1} =f(x_{i+1},y^{(k)}_{i+1} ) . As with the fourth-order Adams–Bashforth–Moulton, this method cannot self start and needs a method such as RK4 for estimating y_1,y_2, and y_3 first.