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MIC3ST 2025 | International Journal of Research and Innovation in Social Science (IJRISS)
Virtual Conference on Melaka International Social Sciences, Science and Technology 2025
ISSN: 2454-6186 | DOI: 10.47772/IJRISS | Special Issue | Volume IX Issue XXIII October 2025
Graded Mesh Number Effect on the Solution of Convection-Diffusion
Flow Problem with Quarter-Circle Source
Aslam Abdullah
*
Department of Aeronautical Engineering, Faculty of Mechanical and Manufacturing Engineering,
Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia
DOI: https://dx.doi.org/10.47772/IJRISS.2025.923MIC3ST250018
Received: 12 August 2025; Accepted: 20 August 2025; Published: 24 October 2025
ABSTRACT
Convection-diffusion equations are fundamental to modeling various transport phenomena in engineering and
scientific applications. However, solving these equations accurately poses significant numerical challenges,
particularly under conditions involving sharp gradients or weak singularities. This study investigates the
influence of graded mesh intervals on the numerical accuracy of a two-dimensional convection-diffusion flow
problem featuring a quarter-circle source. The research focuses on low Peclet number regimes where diffusion
dominates and solution precision is highly sensitive to mesh configuration. The study utilizes a logarithmic
model to generate graded mesh intervals governed by an expansion factor. Sixteen test cases are developed by
applying various mesh spacings to selected Peclet numbers. The numerical solutions are analyzed to quantify
error reduction and assess the convergence behavior across different mesh densities. The results demonstrate a
clear relationship between mesh refinement and solution accuracy, highlighting the graded mesh’s ability to
suppress numerical artifacts such as spurious oscillations and excessive diffusion or dispersion errors. Findings
show that the use of graded meshes significantly enhances the accuracy of scalar concentration profiles,
validating their effectiveness in handling convection-diffusion problems with geometric complexities like
quarter-circle sources. Additionally, the computed orders of accuracy confirm the robustness of the meshing
technique, offering practical insights for optimizing computational resources while maintaining reliability. The
study concludes that selecting appropriate graded mesh parameters—specifically tailored through a
logarithmic model—can serve as a heuristic guide for achieving predictable numerical accuracy in convection-
diffusion simulations. This work contributes to the broader understanding of meshing strategies in
computational fluid dynamics, particularly for low Peclet number applications. It also supports the
development of more efficient and accurate solvers for problems characterized by mixed convective and
diffusive transport, such as the convection-diffusion of water vapor used to describe the dynamics of aircraft
wake vortices.
Keywords: Convection-Diffusion Flow, Graded Mesh, Quarter-Circle Source, Peclet Number, Numerical
Accuracy
INTRODUCTION
Convection-diffusion processes are widely formulated in numerous branches of engineering and physical
sciences, which necessitate a well-designed computational fluid dynamics mesh for obtaining accurate
numerical solutions.
The implementation of a graded mesh plays a crucial role in finite element (FEM) (Kaushik, Kumar, Sharma
& Sharma, 2021; Brdar, Zarin & Teofanov, 2016; Constantinou, Franz, Ludwig & Xenophontos, 2018; Durán,
Lombardi & Prieto, 2013; Chaudhary & Kundaliya, 2022), finite difference (FDM) (Chen, Xu & Zhou, 2019),
exponential B-spline (Kaushik, Kumar, Sharma & Sharma, 2021), and Newton techniques (Chaudhary &
Kundaliya, 2022). Specifically, the mesh proves valuable in numerical investigations of reaction-diffusion
models (Kaushik, Kumar, Sharma & Sharma, 2021; Constantinou, Franz, Ludwig & Xenophontos, 2018;
Durán, Lombardi & Prieto, 2013), singularly perturbed systems with two governing parameters (Brdar, Zarin
& Teofanov, 2016), subdiffusion models incorporating nonlocal diffusion terms (Chaudhary & Kundaliya,
INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN SOCIAL SCIENCE (IJRISS)
ISSN No. 2454-6186 | DOI: 10.47772/IJRISS | Volume IX Issue IX September 2025
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2022), and evolutionary problems characterized by a weakly singular kernel (Chen, Xu & Zhou, 2019). A non-
graded mesh applied to these issues may involve, for example, a local approach (Yedida & Satyanarayana,
2012) to determine an optimal shape parameter for infinitely smooth Radial Basis Functions (RBF) within a
mesh-free framework.
Historically, non-uniform meshing has been extensively explored in the context of solving integro-differential
models. Research has particularly focused on numerical collocation using graded mesh solutions for weakly
singular Volterra integral equations. Likewise, implicit finite difference techniques with non-uniform temporal
steps in time-fractional diffusion models have been actively investigated in recent studies (Chen, Xu & Zhou,
2019).
An improper application of meshing techniques may cause numerical artifacts, such as spurious oscillations,
significant over- or under-predictions, and excessive computational costs. Broad research efforts have been
devoted to different meshing techniques and topologies due to the necessity of solving the equation systems.
Nevertheless, the influence of mesh spacing in graded meshes with an expansion factor
on the accuracy of
two-dimensional convection-diffusion problems with a quarter-circle source under varying Peclet numbers
remains an open problem. This study emphasizes the effect of mesh configurations on solving convection-
diffusion flow models with a quarter-circle source for selected flow parameters. Specifically, the accuracy of
solutions at low Peclet numbers is examined in relation to graded mesh intervals, with the expansion factor
derived from a known logarithmic model of Peclet number established in prior research. The model is
evaluated by assigning multiple graded mesh intervals to each Peclet number of interest, leading to 16 test
scenarios. Quantitative findings establish the accuracy order of the solution to the flow problem. The influence
of graded mesh intervals on solution accuracy thus provides a reference for structured decision-making and
enhances the heuristic approach to selecting computational meshes with a predictable accuracy level,
particularly in computing scalar concentration. Assessing this impact is vital to evaluate the claimed robustness
of graded meshes in resolving the target governing equation. Note that the obtained accuracy orders validate
the concentration profiles.
METHODOLOGY
In differential form, we define general problem model of interest as
for
(1)
where
,
,
, and
are diffusive, convective, reactive, and sink/source terms, respectively,
,
, and
are sufficiently smooth functions, and parameter is unknown. It is assumed that
in
for all
(2)
The reactive and source/sink terms being zero and quarter-circle, respectively, in this paper. Thus for
Eq. (1) is simplified into
(3)
In this work, we explore a model of a convection-diffusion problem with quarter-circle source that is
discretized using finite difference method and solved on graded mesh with various mesh numbers and
expansion factors
. The outcomes of past numerical analysis justify the adoption of the mesh. We observe
average error with respect to mesh number and Peclet number to determine the rate of convergence.
Variation of determines average mesh width.
INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN SOCIAL SCIENCE (IJRISS)
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Generally, the need to solve the system of equations have certainly sparked broad study on various mesh
schemes and structures. The effect of mesh width in graded mesh with mesh expansion factor
on the
solution of 2-dimensional convection-diffusion flow problem with quarter-circle source at various Peclet
numbers , however, is an open question. Examining such effect is essential to challenge the claimed
robustness of graded mesh in solving the governing equation of interest. Quantifying the rate of convergence
of the solution is the aim of this research.
The following are the boundary conditions for the model problem's formulation in Eq. (3):
(4)
In the relevant domain of solution, graded mesh is employed. The interval number is given by
where
an odd integer is the mesh number. In order to define the nodes for the mesh, let us first discretize a defined
independent variable domain in such a way that
The nodes
for the mesh are defined
as
(5)
where
and mesh expansion factor
Clearly
We choose that
41,81
(9)
It was found that the expansion factor
is inversely proportional to the logarithm of the Peclet number ,
for a low Peclet number convection-diffusion flow. The relationship is expressed as
(10)
where
(11)
and
(1
2)
are curve slope and a constant, respectively, in order to systematically set the values of
. In this work, we test
the validity of the relationship in Eq. (10) against a wide range of given in Eq. (9).
The complete method was discussed in details in the previous publication by the author (Abdullah, 2023).
RESULTS
Table 1 presents the numerical errors corresponding to different Peclet numbers (Pe = 3.125, 6.25, 12.5, 25)
across graded meshes with increasing mesh densities (mesh numbers N = 11, 21, 41, and 81). A clear trend is
observed: as the number of mesh intervals increases, the numerical error consistently decreases for all Peclet
numbers. This indicates that finer mesh densities contribute to improved solution accuracy. Additionally, lower
INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN SOCIAL SCIENCE (IJRISS)
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Peclet numbers (e.g., Pe = 3.125) yield smaller numerical errors compared to higher Peclet numbers (e.g., Pe =
25) for the same mesh number, which aligns with the expected diffusion-dominated behavior at low Pe values.
Figure 1 illustrates the scalar concentration profiles θ computed on graded meshes for the four Peclet numbers.
Each subfigure (a) through (d) corresponds to a different Peclet number, increasing from 3.125 to 25. The plots
demonstrate that at low Pe (Figures 1a and 1b), the scalar concentration remains smooth and evenly
distributed, reflecting the dominance of diffusive transport. However, as Pe increases (Figures 1c and 1d), the
scalar field exhibits sharper gradients, particularly near the quarter-circle source. These gradients necessitate
finer mesh resolution to capture the physical behavior without introducing numerical artifacts. The graded
mesh successfully suppresses spurious oscillations and maintains stability, even under increasing convective
effects.
Together, the table and figure highlight the graded mesh’s capability in enhancing numerical precision and
robustness in solving convection-diffusion problems with geometric complexity and varying transport
characteristics.
Table 1: Numerical errors at
Error
3.125
1.1 x 10
-3
2.8 x 10
-4
7.3 x 10
-5
1.8 x 10
-5
6.25
2.2 x 10
-3
3.5 x 10
-4
7.7 x 10
-5
3.8 x 10
-5
12.5
2.8 x 10
-3
5.9 x 10
-4
2.6 x 10
-4
1.3 x 10
-4
25
2.9 x 10
-3
8.4 x 10
-4
4.2 x 10
-4
2.1 x 10
-4
Figure 1: Plot of on graded mesh for (a) (b) (c) (d)
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CONCLUSIONS
This study investigated the influence of graded mesh configurations on the numerical solution of a two-
dimensional convection-diffusion problem with a quarter-circle source. The analysis revealed that refining the
mesh significantly reduces numerical errors, particularly at low Peclet numbers where diffusion dominates.
The application of a logarithmically derived expansion factor to generate graded meshes proved effective in
minimizing solution artifacts and ensuring convergence. Scalar concentration profiles across different Peclet
numbers confirmed the ability of graded meshes to handle both smooth and steep gradients without
compromising accuracy. The results validate the graded mesh approach as a robust and reliable meshing
strategy for convection-diffusion simulations, especially in low Pe regimes. Consequently, selecting
appropriate mesh parameters based on this study can serve as a practical guideline for computational fluid
dynamics simulations involving similar transport phenomena.
ACKNOWLEDGMENTS
This research was supported by Universiti Tun Hussein Onn Malaysia (UTHM) through Tier 1 (vot Q115).
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