Notes on the order of convergence, consistency and stability properties of newly derived schemes

In this paper, two numerical integration methods for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs), namely “Third Order One Step Scheme (TOOSS) and Second Order One Step Scheme (SOOSS)” have been considered. The order of convergence, consistency and the stability properties of the schemes have been investigated. From the analyses, it is observed that SOOSS and TOOSS have second order convergence and third order convergence, respectively. It is also observed that both numerical integration methods are consistent and stable. Moreover, three IVPs of stiff differential equations were solved to examine the performance of SOOSS and TOOSS in terms of absolute relative errors. Hence, the numerical results show that TOOSS performs better than SOOSS because of its higher order of accuracy.


INTRODUCTION
Mathematical model is a means of translating real life situations into mathematical symbols and relations.This concept is commonly used in Sciences and Engineering.Models are developed to help in understanding of physical phenomena.These models frequently resulted in equations that contain derivatives of unknown function of one or several variables.These types of equations are referred to as Differential Equations.Differential equations are not only encountered in physical sciences, but also in diverse fields like Economics, Medicine, Psychology, Operation Research and even in areas such as Biology and Anthropology.In reality, the analytical solutions of some of the equations arising from modelling of real life situations might not be easily obtained.This necessitated the need for approximate solution by the application of numerical methods.To that extent, several algorithms have been proposed in literature based on the nature and the type of the differential equations to be solved such as Wambecq (1976), Jain (2003), Davis (2013), Qureshi et al. (2013), Fadugba and Falodun (2017), Fadugba and Okunlola (2017), just to mention a few.This paper is motivated by the work of Qureshi and Fadugba (2018).In this paper, the order of convergence, consistency and stability properties of the Fadugba-Falodun and Fadugba-Okunlola schemes were investigated.The TOOSS and SOOSS have been applied on IVPs of first and second order ODEs (Fadugba and Ajayi, 2017;Qureshi and Fadugba, 2018;Fadugba, 2019;Fadugba, 2020).

ORDER OF CONVERGENCE OF THE SCHEMES
In order to check the order of convergence of the schemes, the formula of the schemes is subtracted from the Taylor's series expansion for y(x) in powers of h under the localizing assumptions.The convergence of TOOSS and SOOSS is presented as follows.

Order of convergence of TOOSS
From the Taylor's series, one obtains: Fadugba and Okunlola (2017) derived a scheme of the form: Using Equations 1 and 2, the local truncation error becomes: Replacing the term h e 2 in Equation 4 by its Maclaurin's series given by .
(5) Therefore, Equation 4 becomes: Under the local assumptions, the terms up to h 3 have been cancelled, then Equation 7 becomes: Hence, the TOOSS has the convergence of third order.

Order of convergence of SOOSS
The Taylor's series expression for ) (x y in powers of h is given by: Fadugba and Falodun (2017) derived a scheme of the form: The local truncation error is given by Using Equations 9 and 10, Equation 11 becomes: Using the Maclaurin's series expansion of h e  and simplifying the Equation 12 under the localizing assumption, one gets: Hence, SOOSS has the convergence of second order.

Remark
The analysis of local truncation error indeed determines the order of convergence for any numerical technique designed to solve IVPs in ODEs.

CONSISTENCY PROPERTIES OF THE SCHEMES
It is a known fact that any numerical method having an order of accuracy greater than or equal to 1 is considered to be consistent.In other words, a numerical integration method is said to be consistent if it has at least order p =1.

Consistency analysis of TOOSS
Among many, one of the ways to analyze the consistency of a numerical technique is to check that whether (Qureshi and Fadugba, 2018): From Equation 14, it is observed that TOOSS has consistency characteristics.

Consistency analysis of SOOSS
Following the same procedures as that of the Qureshi and Fadugba (2018), one obtains that: From Equation 15, it is observed that SOOSS has consistency characteristics.

Remark
For a numerical technique to be consistent, it is important Fadugba 3 for the truncation errors to be zero when the step size h gets smaller and ultimately reaches to zero.

STABILITY ANALYSES OF THE SCHEMES
A numerical integration method is said to be stable if it is capable of damping out the small fluctuations carried out in the input data.A one step explicit numerical integration method is reserved to be stable if a small perturbation in the initial conditions of the IVP leads to a small perturbation in the following numerical approximation.

Stability analysis of TOOSS
For the stability analysis of TOOSS, one of the popular ways is to apply the scheme to the Dahlquist's test equation: whose exact solution is given by where The exact solution at the point The numerical approximation obtained using TOOSS gives Using Equation 24, the stability region is plotted in Figure 1.Hence, TOOSS is found to be stable in Figure 1.

Stability analysis of SOOSS
To discuss the stability analysis of SOOSS, consider the following Dahlquist's test equation of the form: The exact solution of Equation 25 is given by and following the procedures of Qureshi and Fadugba (2018); the exact solution at the point When applied SOOSS on this test problem, one gets: Comparing Equations 27 and 28, it is clearly seen that Equation 28is a three-term approximation for the function h e  in the exact solution.The error growth factor given by Equation 28 can be controlled by 1  B so that the errors may not amplify.Thus, the stability function of SOOSS requires that: Therefore, Equation 30 becomes: The region of absolute stability for SOOSS is defined by the region in the complex plane such that . The stability region is plotted in Figure 2.

Remark
The notion of stability may be taken in different contexts: it may be associated with the specific numerical technique used, or the step size h used in numerical computations or with the particular problem being solved.

NUMERICAL EXAMPLES AND DISCUSSION
Here presents the implementation of the two schemes on IVPs of stiff differential equations.The discussion of     0.00000000 0.00000000 10 -4 , 10 -5 , 10 -6 .It is clearly seen in Tables 1 to 3 for every decrease in h, there are second order and third-order decrease in the magnitude of the computed FABRE via SOOSS and TOOSS, respectively.It is observed from the Figures 3 to 5 that the FABREs generated via SOOSS are greater than that of the TOOSS.

Conclusion
In this paper, notes on the order of convergence, consistency and stability properties of TOOSS and SOOSS have been successfully presented.It is observed that TOOSS and SOOSS have the convergence of third order and second order, respectively.From the analysis, it is observed that the two methods are convergent, consistent since they have order of accuracy greater than 1.It is also observed that TOOSS and SOOSS are stable as shown in Figures 1 and 2, respectively.Hence, it can be concluded from the numerical results that TOOSS performs better than SOOSS since it has a higher order of accuracy.3.

CONFLICT OF INTERESTS
The author has not declared any conflict of interests.

Table 1 .
The FABRE via SOOSS and TOOSS for Example 1.

Table 2 .
The FABRE via SOOSS and TOOSS for Example 2.

Table 3 .
The FABRE via SOOSS and TOOSS for Example 3.