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Global Research journal of Natural Science  
& Technology (GRJNST)  
Volume: 04 - Issue 2 (2026), 2079  
ISSN P: 2790-7643 ISSN E: 2790-7651  
www.grjnst.net  
https://doi.org/10.53762/grjnst.04.02.30  
A Theoretical Investigation of Local Truncation Error and Convergence of  
an Accelerated Third-Order Runge–Kutta Method for Non-Autonomous  
Ordinary Differential Equations  
Received: 29 March 2026. Accepted: 22 April 2026. Published: 30 April 2026  
Muhammad Daud Kandhro(Corresponding Author)  
Institute of Mathematics and Computer Science, University of Sindh, Jamshoro  
Subject Specialist (BPS-17) in School Educationand Literacy Departmen (SELD), Govt. of Sindh  
daud.kandhro@gmail.com  
Naveed Ahmed Tunio  
HST (BPS-16) in School Education and Literacy Department (SELD), Govt. of Sindh  
naveedtunio@gmail.com  
Zohaib Ali Qureshi  
Institute of Mathematics and Computer Science, University of Sindh, Jamshoro  
zohaib.ali@usindh.edu.pk  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
https://doi.org/10.53762/grjnst.04.02.30  
Copyright © 2026 GRJNST. This article is published under an Open Access model. It is made available to the public under the terms of the Creative  
Commons Attribution 4.0 International (CC BY 4.0) license, which permits unrestricted use and distribution  
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Abstract:  
In this article, we theoretically examined well-organized third order numerical  
technique for IVP of ODE’s including partial derivative which has enhanced its  
competency regarding truncation error. Accelerated numerical method for local  
truncation error and convergence is theoretically investigated to assess how  
accurate and reliable proposed method is. Expansion of Taylor’s series is done  
that provides right pattern to expand and evaluate function evaluation. With the  
help of Taylor’s series, Local truncation error is an explicitly derived to clarify  
the order of accuracy. Linear standard test is discussed for calculating stability.  
Stability is investigated for knowing behavior of method. Stability region is  
drawn by MATLAB2023a software. Stability region is visualized to check  
numerical method possess a bounded solution when applied frequently and  
repeatedly. Consistency is proved which tells us error goes to zero as much as  
we decrease step size which guarantees the desired result. Convergence criteria  
theoretically discussed here by proving consistency and stability. Theoretically  
findings showing that the numerical method is stable, attains high accuracy and  
gives reliable performance. Therefore, method is efficient and applicable for  
extensive class of IVP arising in the area of ODE’s.  
Keywords: Numerical Method, Local Truncation Error, Stability, Consistency,  
Convergence  
1. INTRODUCTION  
Ordinary differential equations (ODEs) take very crucial role in the formation of  
modeling arising in science, engineering, physics, biology, and economics. Many real-  
world problems are governed by initial value problems whose analytical solutions are  
either difficult or impossible to obtain in closed form. As a result, numerical methods  
have become indispensable tools for approximating solutions of such equations. When  
designing numerical schemes, computational efficiency is not the only consideration for  
obtaining reliable results; it's also important to understand the theoretical aspects of the  
schemes, such as their accuracy and stability.  
One of the key elements in the analysis of numerical schemes is error analysis. The  
analysis of error helps determine the impact of discretization on the numerical solution  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
https://doi.org/10.53762/grjnst.04.02.30  
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and enables the comparison between different numerical schemes. In these studies, it is  
common to refer to local and global errors.  
The local truncation error (LTE) is defined as the error presented in a single step of the  
numerical method, assuming that the earlier steps are exact. It serves as a primary  
measure for determining the order of accuracy. Specifically, if the LTE behaves like  
then the method has order p. Thus, the derivation and analysis of the LTE are  
essential for constructing of high-accuracy numerical methods.  
Closely linked to error analysis is the concept of consistency, which guarantees that the  
consequence approximate technique precisely exemplifies the underlying differential  
equation as step size nearer to zero. Consistency guarantees that the discrete scheme is a  
faithful approximation of the continuous problem, forming a necessary condition for  
convergence. However, consistency alone is not sufficient to ensure reliable numerical  
results.  
Another important property is stability, relating to the behavior of numerical errors in  
the iterative process. Stability is a property that guarantees that errors (due to truncation,  
round-off or small changes in initial conditions) do not blow up during the iterative  
process. This is crucial for stiff problems and when integrating over long times. A very  
popular method for investigating stability involves application of a standard linear test  
equation. Using such an approach, one establishes the so-called stability function, which  
describes the growth in error in each step.  
The stability region is introduced as a part of this process. The region in the complex  
plane for which the numerical solution is bounded is known as the stability region. The  
extent and shape of the stability region give an idea of the step sizes that can be taken  
and the stability of the method. It is preferable for a method to have a larger stability  
region, particularly for stiff or highly oscillatory problems, because it means that larger  
step sizes can be taken.  
The main requirement of method is that it is convergent, which means that the  
numerical solution converges to the exact solution as the step size approaches zero.  
Convergence is a crucial property for a numerical method. A fundamental result in  
numerical analysis is that, under certain conditions, consistency and stability lead to  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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convergence. This connection underscores the need to establish both of these properties  
of a numerical method.  
In light of these considerations, the development of effective and reliable numerical  
schemes requires a comprehensive framework that integrates error analysis, consistency,  
stability, and convergence. In this paper, we focus on the systematic investigation of  
these properties for a method designed to solve ODE’s. The analysis contains the  
derivation of the LTE to establish the order of accuracy, verification of consistency,  
detailed stability analysis through the construction of the stability function, and  
characterization of the corresponding stability region. Finally, convergence is established  
by combining the results of consistency and stability.  
2. MATERIALS AND METHODS  
Many numerical methods have designed to attain the estimated consequences of IVP’s  
for differential equation in nature ordinary. We can consider general function form:  
Interval we have given as  
is parted into a uniform parts,  
, the step size is  
equidistant node  
. The to solve the function in a discrete series  
to attend approximate values  
.
Generally, function of two space variable also familiar as a derivative of the dependent  
variable regarding to independent has been calculated through integrating  
Many equations named differential don’t reach to the solution such as particular and  
analytical. To deal with such difficulty new innovations take place in numerical analysis.  
Best advantage of numerical scheme is that these have better performance than analytical.  
Researchers day by day are trying to innovate many methods to obtain better  
consequences. But still huge number of work is needed to understand proper behaves of  
method. Researchers and scholars are out on the field of innovation to hone their skills.  
Many of them have also done a great job to construct and modified new methods.  
Kandhro [4] has developed accelerated method whose Iterative Integrator which is  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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This is newly developed an accelerated explicit scheme having three function evaluations  
per time step. Now error analysis.  
3. ERROR ANALYSIS  
It is main purpose to solve ordinary differential equation numerically to attain results  
which are as close as possible to the exact solution. Two sources of error which affect  
the accuracy of numerical method named as round-off (RO) and truncation (TR). RO  
round-off error takes the place when computers can only stock numbers with limited  
precision and TR arises because mathematical procedures are approximated (e.g., by  
deserting higher-order terms). An accuracy can totally rely on blunder what size of step  
size is taken. We consider Taylor’s series for two variable up to the fourth power of step  
size , we have  
The local truncation error is an error which spawned in a unique step of the proposed  
improved scheme that is documented as  
where  
Where  
is the solution obtained by Taylors’s Series and  
is used as an  
and similar  
approximate solution. Taylor series is utilized to expand these around  
terms collect in . Now we expand proposed accelerated explicit Method present in eqn  
(2) upto , we get  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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Subtract (5) from (3). The proposed scheme has a local truncation error that is:  
The local truncation error (LTE) often calls with name "discretization error per time  
step", which helps to evaluate the order of accuracy. From LTE it is observed, LTE has  
4rth order then the proposed scheme in [4] has one less than the LTE in order.  
Therefore, the proposed scheme in [4] has third order of accuracy.  
4. CONSISTENCY ANALYSIS  
Definition 5.1 An increment function  
be consistent, when  
of numerical method is entitled to  
Consistency of the numerical methods tells local truncation error (LTE) tends to zero as  
the step size decreases or  
function as  
. Therefore, newly proposed scheme has increment  
By utilizing lim both sides  
h0  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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Hence, the newly proposed scheme with at least third order accuracy is proven to be  
consistent.  
5. LINEAR STABILITY ANALYSIS  
Dahlquist’s test problem is to be considered for verifying the stability of the Method in  
[4]  
whereas  
where is a complex constant i.e,  
. After executing Method (2) on this test, we  
attain polynomial function acknowledged as stability function with linear region which  
displayed via unfilled area in fig 1.  
Substitute all values in (5), polynomial form of stability function is derived as  
where  
This is Stability Function for accelerated proposed method in [4] which reflects the  
third-order accuracy of the method. When errors are introduced then linear stability is  
checked to know how numerical method behaves. In particular, it expresses that whether  
those errors decay, remain bounded, or grow uncontrollably during the computation.  
Stability controls error propagation.  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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6. REGION OF ABSOLUTE STABILITY  
The region of the accelerated proposed method [4] is computed and visualized using  
MATLAB. A grid of complex values is generated, and the stability function  
is evaluated over this domain. The set of points satisfying  
is identified  
and plotted to illustrate the stability region in the complex plane.  
This stability region is a fundamental concept used to determine whether a numerical  
method produces bounded solutions when applied repeatedly. Figure 1 illustrates the  
two-dimensional region of absolute stability for the proposed numerical method in the  
complex plane. The real part  
is demonstrated on horizontal line of axis , while  
the imaginary part is demonstrated on vertical axis of line, where  
. The  
white (unshaded) regions indicate the set of values for which the stability condition  
is satisfied. These regions represent the domain when solution approximately  
remains stable and keep bounded. The orange (shaded) region corresponds to values of  
GRJNST, Volume: 04 - Issue 2 (2026) / ISSN P: 2790-7643  
Article ID: 2079  
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for which  
indicating instability, where numerical errors may grow  
exponentially.  
7. CONVERGENCE OF THE METHOD  
A method is to be numerically convergent when both consistency and stability justified.  
The necessary condition for convergence of a numerical method is consistency, which  
guarantees that the local discretization error vanishes as step-size very nearly approaches  
zero. However, convergence is achieved only when consistency is combined with  
stability. Therefore, above said words the accelerated proposed method is converges.  
8. CONCLUSION  
In this work, the construction and detailed mathematical analysis of a numerical method  
for solving initial value problems of ordinary differential equations. we theoretically  
examined well-organized third order numerical technique for IVP of ODE’s including  
partial derivative which has enhanced its competency regarding truncation error. The  
main focus is not here to derive scheme but here is justify its theoretically behave which  
is analyzed by systematic investigation of error and convergence. The LTE has derived  
perfectly which endorsing accuracy by identify its order. Stability is investigated for  
knowing behavior of method. Stability region is envisioned to check numerical method  
possess a bounded solution when applied frequently and repeatedly. Convergence criteria  
theoretically discussed here by proving consistency and stability. Theoretically findings  
showing that the numerical method is stable, attains high accuracy and gives reliable  
performance. Therefore, method is efficient and applicable for extensive class of IVP  
arising in the area of ODE’s.  
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