Proficiency of Second Derivative Schemes for the Numerical Solution of Stiff Systems

This paper presents a study on the development and implementation of a second derivative method for the solution of stiff first order initial value problems of ordinary differential equations using method of interpolation and collocation of polynomial approximate solution. The results of this paper bring some useful information. The constructed methods are A-stable up to order 8. As it is shown in the numerical examples, the new methods are superior for stiff systems.


Introduction
We considered development of second derivative method for the solution of ( ) ( ) 0 , , , where n x is the initial points, [ ] is continuous and at least twice differentiable.We seek the solution on equidistant set of points defined on the integration interval , N is a positive integer.
A potentially good numerical method for the solution of stiff systems must have good accuracy and reasonably wide region of absolute stability.A-Stability requirement is the minimum criteria on the choice of suitable methods.The search for higher order A-stable linear multistep method is carried out in two ways; firstly, the use of higher derivatives of the approximate solution and secondly, the inclusion of additional stages of off grid points Ezzeddine and Hojjati [1].
The aim of this paper is to develop a class of second derivative linear multistep method with varying step-lengths which are A-stable with large region of absolute stability (see Figures 1-3).The three methods recovered are tested on some numerical examples and their results compared with each other in order to determine how to fix the varying step-lengths to obtain the best results as shown Tables 1-4.

Development of the Method
We considered the approximate solution of the form ( ) where n a 's are constants to be determined.The ith derivative of (2) gives Imposing the following conditions on (2)   where ( ) where Solving the system of nonlinear equations using Newton Raphson's method gives ( ) ( ) where J κ is the Jacobian matrix.The necessary and sufficient condition for convergence of ( 8) is that the spectral radius of the inverse of the Jacobian matrix ( )

Specification of the Method
In this paper, we consider grid points , 0 , , , ,

Stability Properties
In this section, we investigate the basic properties of the developed method vis-a-vis order, local truncation error, consistency, zero-stability, convergence, and region of absolute stability of the methods.

Order of Convergence
The operation  is associated with the linear method defined by ( ) where ( ) y x is an arbitrary function, continuously differentiable on an interval [ ] , n N x x .Ehigie, J. O. and Okunuga [11] can be written in Taylor expansion as where ( )

Consistency
Definition 1 A block method is consistent if it has order 1. p ≥

Zero-Stability
Definition 2 A method is said to be Zero-stable if no root of the first characteristics polynomial has modulus greater than one, and if every root of modulus one has multiplicity not greater than one or is simple.

Convergence
Definition 3 A method is said to be convergent if It is consistent and zero stable.

Region of Absolute Stability (RAS)
Within the range 0 20 x ≤ ≤ .The eigenvalues of the Jacobian matrix

Conclusion
In this paper, we introduced three new second derivative linear multistep methods for the numerical solution of stiff initial value problems.Four numerical examples were considered, the results justified the proficiency of the second derivative method which is cheaper to implement since it does not require starting values and particularly the new methods show that lower step method gives better accuracy than higher step methods of the same order.We are able to achieve the aim of this paper which is to develop a class of second derivative linear multistep methods that are A-stable with large region of absolute stabilityas shown in Figures 1-3.The results from the high-order methods are very encourging (see Tables 1-4), therefore, we recommend further investigation of the second-derivative.

Figure 3 .
Figure 3. Showing RAS for Case III.
For our methods, in case 1, we considered one step method with two hybrid points with equal interval where 1 .For case II, we considered two step method with one hybrid points with equal interval, where

Definition 4 The
Region of absolute stability (RAS) of a L M M is the set R = { z h λ = : for z where the root of the stability polynomial are absolute less than one}.

Table 1 .
Showing Results of Example 1.

Table 2 .
Showing Results for Example 2.

Table 3 .
Showing Results for Example 3.

Table 4 .
Showing Results for Example 4.
called the error constant and