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Figure 2-4 illustrate the region of absolute stability for first-order Huen's method while Figure 5-7 exhibit the regions of absolute stability for second-order Huen's method. The stability region is inside the boundary of the dotted lines. As noticeable that the stability region is larger when the step size is half (w=1/2) followed by when the step size being constant (w=1) and the smallest stability region is determined when the step size is double (w=2). As conclusion, the stability region is getting improve as the step size getting smaller.

Mister Micawber · Accepted Answer

Figures 2-4 illustrate the region of absolute stability for first-order Huen's method while Figures 5-7 exhibit the regions of absolute stability for second-order Huen's method. The stability region is inside the boundary of the dotted lines. Notice that the stability region is largest when the step size is half (w=1/2) and smaller when the step size is constant (w=1), with the smallest stability region determined by a double step size (w=2). In summary, the stability region grows as the step size becomes smaller.

Myvi · Answer

Sir, since in another chapter in my thesis, I need to discuss the stability region for another method and the pattern is similar as previous chapter. So, can I just simply mention that the pattern of stability region of DEF method are same as ABC method which it grows as step size becomes smaller..

Thanks a a lot..

Mister Micawber · Answer

I suppose so, but I would like to see the precise sentences you want to use.

Myvi · Answer

Sorry for late response sir..Your contribution is greatly appreciated.

1- As mentioned earlier in chapter two, the analysation of the stability regions play a big role in determining a suitable method for a particular problem. Hence, this section will discuss the stability region of the GHI method for first and second order BVPs such use the steps applied in chapter two as well. Again the test equation (2.3) is employing into the first-order corrector formulae and then that formulae are representing in a matrix form.

2- Following the similar steps as w=2 for w=3 and w=4 and then solving the determinant of F=p+3w with p=2h, getting the stability polynomials as follows

3- The stability region obtained according to the stability polynomials (4.5, 4.6, 4.7) can be seen in Figure 23, 24 and 25.

4- In a similar manner as chapter two, the test equation (2.4) is employed to the corrector formulae of second-order GHI method with showing the obtained formulae as follows

5- Finally, solving the determinant of F=p+q+3w with p=2h and q=3h and the resulting stability polynomial is given as

6- The stability regions are defined by these stability polynomials (4.8, 4.9, 4.10) are shown as follows

In summary, the stability regions for first- and second-order GHI method are shown as Figure (23, 24, 25) and (26, 27, 28) respectively and it lies insides the dotted line contour. Looking at these stability regions obtained, as the step size becomes smaller, the size of stability regions increased. This pattern is happened as the earlier methods introduced which are ABC and DEF method.

Mister Micawber · Answer

You overuse 'hence'.

As mentioned earlier in Chapter 2, the analysis of the stability regions play a big role in determining a suitable method for a particular problem. This section will discuss the stability region of the GHI method for first- and second-order BVPs, which use the steps applied in Chapter 2 as well. Again the test equation (2.3) is employed in the first-order corrector formulae and then those formulae are represented in a matrix form.

Following similar steps as w=2 for w=3 and w=4 and then solving the determinant of F=p+3w with p=2h, we obtain the stability polynomials as follows.

The stability region obtained according to the stability polynomials (4.5, 4.6, 4.7) can be seen in Figures 23, 24 and 25.

In a similar manner as in Chapter 2, the test equation (2.4) is employed for the corrector formulae of the second-order GHI method, producing the obtained formulae as follows.

Finally, solving the determinant of F=p+q+3w with p=2h and q=3h and the resulting stability polynomial is given as

The stability regions defined by these stability polynomials (4.8, 4.9, 4.10) are shown as follows

In summary, the stability regions for the first- and second-order GHI method are shown in Figures 23, 24, 25 and Figures 26, 27, 28 respectively: they lie within the dotted line contour. Looking at these stability regions obtained, as the step size becomes smaller, the size of the stability regions increases. This pattern occurs also in the ABC and DEF methods introduced earlier.

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