INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025  
Impact of the Peclet Number on Numerical Solutions of Modified  
Convection-Diffusion Equations Using Berger's Equation  
T M A K Azad  
Associate professor, Department of Computer Science & Engineering, University of Liberal Arts  
Bangladesh, 288 Beribadh Road, Mohammadpur, Dhaka-1207.  
DOI : https://doi.org/10.51583/IJLTEMAS.2025.1412000019  
Received: 13 December 2025; Accepted: 20 December 2025; Published: 27 December 2025  
ABSTRACT:  
The study investigates the influence of the Peclet number on the numerical solutions of modified convection-  
diffusion equations, specifically utilizing Berger's equation as the governing model. Finite difference methods  
has been employed to solve Convection diffusion equation under different Peclet numbers, analyzing the  
resulting numerical behavior, including solution profiles, error propagation, and computational efficiency. The  
findings reveal that as the Peclet number increases, the dominance of convective terms introduces numerical  
instabilities, such as oscillations and excessive diffusion, necessitating the implementation of specialized  
discretization techniques or stabilization methods. The study also explores the effectiveness of upwind schemes  
and adaptive mesh refinement in mitigating these challenges.  
Keywords: Peclet number, numerical solution, modified Convection-Diffusion Equation, Burger’s Equation,  
Finite Difference Schemes, and Stability Conditions.  
INTRODUCTION:  
The study of convection-diffusion equations is fundamental in various fields of science and engineering,  
including fluid dynamics, heat transfer, and environmental modeling. These equations describe the transport of  
physical quantities, such as mass, heat, or momentum, due to the combined effects of convection (advection)  
and diffusion. The Peclet number (Pe), a dimensionless parameter, plays a critical role in determining the relative  
importance of these two processes. It is defined as the ratio of convective to diffusive transport rates and is given  
by:  
퐶표푛푣푒푐푡푖표푛 푡푟푎푛푠푝표푟푡 푟푎푡푒  
퐷푖푓푓푢푠푖표푛 푟푎푡푒  
푢퐿  
퐷
푃푒 =  
=
where u is the characteristic velocity, L is the characteristic length scale, and D is the diffusivity. High Peclet  
numbers indicate dominance of convection, while low Peclet numbers signify diffusion-dominated regimes.  
In many practical applications, the standard convection-diffusion equation is modified to account for additional  
physical phenomena, such as source terms, nonlinearities, or boundary layer effects. Berger's equation, a specific  
form of the modified convection-diffusion equation, has been widely used to model such scenarios. However,  
the numerical solution of these equations is often challenging, particularly when the Peclet number is large, as  
it can lead to numerical instabilities, oscillations, or excessive diffusion if not handled properly.  
The impact of the Peclet number on numerical solutions has been extensively studied in the context of standard  
convection-diffusion equations [1], [2], [3]. However, its influence on modified equations, such as Berger's  
equation, remains less explored. Understanding this relationship is crucial for developing robust numerical  
methods that can accurately capture the physical behavior of the system across a wide range of Peclet numbers.  
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Many researchers have already been working on it.  
Changjun and Shuwen [3] made a numerical simulation on river water pollution by using grey differential model.  
They corrected the model in finding the truncation error and found that the obtained results from the grey model  
are excellent and reasonable.  
Atul Kumar, Dilip Kumar Jaiswal and Naveen Kumar [5] presented an analytical solution of the one-dimensional  
ADE by reducing the original ADE into a diffusion equation by using Laplace transformation technique. Mehdi  
Dehghan [6] presented the solution of the one-dimensional convection diffusion equation with constant  
coefficient by using several finite difference schemes.  
This work investigates the effect of the Peclet number on the numerical solutions of modified convection-  
diffusion equations using Berger's equation as a case study. By analyzing the performance of various numerical  
schemes under different Peclet regimes, we aim to provide insights into the challenges and strategies for  
achieving accurate and stable solutions. The findings are expected to contribute to the development of more  
efficient numerical methods for solving complex convection-diffusion problems in engineering and scientific  
applications.  
Governing Equation and Numerical Schemes  
Governing Equation  
In this paper, we consider variable advection velocity u(t, x), so that the PDE reads as convection-diffusion  
2
equation (CDE) 휕ꢀ + 푢 휕ꢀ = 퐷 휕 ꢀ where, we have two unknowns c(t, x) and u(t, x). Therefore, we have to solve  
2
휕ꢁ  
휕푥  
휕푥  
2
휕 ꢂ  
another equation and we select the viscous Burger’s equation 휕ꢂ + 푢 휕ꢂ =   
to compute the variable  
2
휕ꢁ  
휕푥  
휕푥  
velocity u(t, x). Our problem is thus to solve the following system of PDE’s simultaneously as an IBVP  
2
휕ꢂ  
휕ꢂ + 푢  
휕ꢁ  
= 휕 ꢂ  
,
,
푎 < ꢃ < 푏,  
푎 < ꢃ < 푏,  
푡 > 0,  
푡 > 0,  
(1a)  
(1b)  
2
휕푥  
휕푥  
2
휕ꢀ + 푢 휕ꢀ = 퐷 휕 ꢀ  
2
휕ꢁ  
휕푥  
휕푥  
Appended with initial condition  
(
)
( )  
(
)
( )  
푢 ꢃ, 0 = 푓 ꢃ ;  
푐 ꢃ, 0 = 푓 ꢃ  
푎 ≤ ꢃ < 푏  
and Neumann boundary conditions  
ꢄ
ꢄ
(
)
( )  
(
)
( )  
푢 푡, 푎 = 푢ꢅ 푡 ;  
푢 푡, 푏 = 푢ꢆ 푡  
푡0 ≤ 푡 ≤ 푇  
ꢄꢃ  
ꢄꢃ  
ꢄ
ꢄ
(
)
( )  
(
)
( )  
푐 푡, 푎 = 푐ꢅ 푡 ;  
푐 푡, 푏 = 푐ꢆ 푡  
푡0 ≤ 푡 ≤ 푇  
ꢄꢃ  
ꢄꢃ  
where ca, cb, ua, ub are constant concentration values.  
Explicit Upwind Difference Scheme (FTBSCS)  
We consider our second order system of PDE’s simultaneously  
2
휕ꢂ + 푢 휕ꢂ = 휕 ꢂ  
,
(2a)  
2
휕ꢁ  
휕푥  
휕푥  
2
휕ꢀ  
휕푥  
휕 ꢀ  
휕ꢀ + 푢  
휕ꢁ  
= 퐷  
(2b)  
2
휕푥  
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ꢈ
ꢈ
(
)
Let the solution 푢 ꢃꢇ, 푡ꢈ be denoted by 푈ꢇ and its approximate value by 푢ꢇ .  
휕ꢂ  
The discretization of  
is obtained by first order forward difference in time  
휕ꢁ  
ꢄ푢  
ꢄ푡  
푈ꢇꢈ+1 − 푈ꢇꢈ  
∆푡  
≈
+ 푂(∆푡)  
휕ꢂ  
휕푥  
The discretization of  
is obtained by first order backward difference in space  
ꢄ푢  
ꢄꢃ  
푈ꢇꢈ − 푈ꢇꢈ−1  
∆ꢃ  
≈
+ 푂(∆ꢃ)  
2
휕 ꢂ  
Discretization of  
is obtain from second order centered difference in space  
2
휕푥  
ꢄ2푢  
ꢄꢃ2  
푈ꢇꢈ−1 − 2푈ꢇꢈ + 푈ꢇꢈ+1  
2
(
)
≈
+ 푂 ∆ꢃ  
∆ꢃ2  
The simplest numerical discretization of (2a) is  
푢ꢇꢈ+1 − 푢ꢇꢈ  
푢ꢇꢈ − 푢ꢇꢈ−1  
∆ꢃ  
푢ꢇꢈ+1 − 2푢ꢇꢈ + 푢ꢇꢈ−1  
+ 푢ꢇꢈ  
∆푡  
=   
,
∆ꢃ2  
ꢈ
ꢈ
ꢈ
푢ꢇꢈ+1 = γ + 푝푒 푢ꢇ−1 + 1 − 훾 − 2pe 푢ꢇ + pe푢ꢇ+1  
,
(3a)  
(
)
(
)
We get,  
∆푡  
∆푡  
where, 훾 =  
ꢈ
푢ꢇꢈ ,  
푟 = 푝푒 =  
∆ꢃ  
∆ꢃ2  
ꢈ
(
)
Let the solution 푐 ꢃꢇ, 푡ꢈ be denoted by 퐶ꢇ and its approximate value by 푐ꢇ .  
휕ꢉ  
The discretization of  
is obtained by first order forward difference in time  
휕ꢁ  
ꢄ푐  
ꢄ푡  
퐶ꢇꢈ+1 − 퐶ꢇꢈ  
∆푡  
≈
+ 푂(∆푡)  
The discretization of 휕ꢉ is obtained by first order backward difference in space  
휕푥  
ꢄ푐  
ꢄꢃ  
퐶ꢇꢈ − 퐶ꢇꢈ−1  
∆ꢃ  
≈
+ 푂(∆ꢃ)  
2
휕 ꢉ  
Discretization of  
is obtain from second order centered difference in space  
2
휕푥  
ꢄ2푐  
ꢄꢃ2  
퐶ꢇꢈ−1 − 2퐶ꢇꢈ + 퐶ꢇꢈ+1  
2
(
)
≈
+ 푂 ∆ꢃ  
∆ꢃ2  
The simplest numerical discretization of (2b) is  
푐ꢇꢈ+1 − 푐ꢇꢈ  
푐ꢇꢈ − 푐ꢇꢈ−1  
∆ꢃ  
푐ꢇꢈ+1 − 2푐ꢇꢈ + 푐ꢇꢈ−1  
+ 푢ꢇꢈ  
∆푡  
= 퐷  
,
∆ꢃ2  
ꢈ
ꢈ
ꢈ
we get, 푐ꢇꢈ+1 = 훾 + 푝푒 푐ꢇ−1 + 1 − 훾 − 2푝푒푐ꢇ + 푝푒푐ꢇ+1  
,
(3b)  
(
)
(
)
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∆푡  
퐷∆푡  
∆ꢃ2  
where, 훾 =  
푢ꢇꢈ ,  
= 푝푒 =  
∆ꢃ  
Which is the explicit upwind difference scheme and it is also known as FTBSCS techniques. The stability  
condition is controlled by  
∆ꢁ  
∆ꢁ  
ꢊ∆ꢁ  
훾 =  
푢ꢇꢈ , 푟 = 푝푒 =  
and 훾 = ∆ꢁ 푢ꢇꢈ , = 푝푒 =  
2
2
∆푥  
∆푥  
∆푥  
∆푥  
It is seen that the truncation errors for the forward and backward differences are of first order; whereas the  
centered difference yields a second order truncation error (using by Taylor Series expansions). Therefore, the  
scheme outlined above is consistent.  
Explicit Centered Difference Scheme (FTCS)  
We consider our second order system of PDE’s simultaneously  
2
휕ꢂ  
휕ꢂ + 푢  
휕ꢁ  
= 휕 ꢂ  
,
(4a)  
(4b)  
2
휕푥  
휕푥  
2
휕ꢀ  
휕푥  
휕 ꢀ  
휕ꢀ + 푢  
휕ꢁ  
= 퐷  
2
휕푥  
ꢈ
ꢈ
(
)
Let the solution 푢 ꢃꢇ, 푡ꢈ be denoted by 푈ꢇ and its approximate value by 푢ꢇ .  
The discretization of 휕ꢂ is obtained by first order forward difference in time  
휕ꢁ  
ꢄ푢  
ꢄ푡  
푈ꢇꢈ+1 − 푈ꢇꢈ  
∆푡  
≈
+ 푂(∆푡)  
휕ꢂ  
휕푥  
The discretization of  
is obtained by first order centered difference in space  
ꢄ푢  
ꢄꢃ  
푈ꢇꢈ+1 − 푈ꢇꢈ−1  
2
(
)
≈
+ 푂 ∆ꢃ  
2∆ꢃ  
2
Discretization of 휕 ꢂ is obtain from second order centered difference in space  
2
휕푥  
ꢄ2푢  
ꢄꢃ2  
푈ꢇꢈ−1 − 2푈ꢇꢈ + 푈ꢇꢈ+1  
2
(
)
≈
+ 푂 ∆ꢃ  
∆ꢃ2  
The simplest numerical discretization of (4a) is  
푢ꢇꢈ+1 − 푢ꢇꢈ  
∆푡  
푢ꢇꢈ+1 − 푢ꢇꢈ−1  
푢ꢇꢈ+1 − 2푢ꢇꢈ + 푢ꢇꢈ−1  
+ 푢ꢇꢈ  
=   
,
2∆ꢃ  
∆ꢃ2  
∆ꢁ  
푢ꢇꢈ+1 = 푢ꢇꢈ − 2∆푥 푢ꢇꢈ 푢ꢇ+1 − 푢ꢇ−1  
+
∆ꢁ (푢ꢇꢈ+1 − 2푢ꢇꢈ + 푢ꢇꢈ−1),  
ꢈ
ꢈ
(
)
2
∆푥  
푢ꢇꢈ+1 = 푢ꢇꢈ − ꢋ 푢ꢇ+1 − 푢ꢇ−1 + 푟(푢ꢇ+1 − 2푢ꢇꢈ + 푢ꢇ−1),  
ꢈ
ꢈ
ꢈ
ꢈ
(
)
2
ꢋ
ꢈ
ꢋ
ꢈ
ꢈ
we get, 푢ꢇꢈ+1 = (푝푒 + 2)푢ꢇ−1 + 1 − 2푝푒 푢ꢇ + (pe − 2) 푢ꢇ+1  
,
(5a)  
(
)
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∆푡  
∆푡  
∆ꢃ2  
where, 훾 =  
푢ꢇꢈ ,  
푟 = 푝푒 =  
∆ꢃ  
ꢈ
ꢈ
(
)
Let the solution 푐 ꢃꢇ, 푡ꢈ be denoted by 퐶ꢇ and its approximate value by 푐ꢇ .  
The discretization of 휕ꢀ is obtained by first order forward difference in time  
휕ꢁ  
ꢄ푐  
ꢄ푡  
퐶ꢇꢈ+1 − 퐶ꢇꢈ  
∆푡  
≈
+ 푂(∆푡)  
The discretization of 휕ꢀ is obtained by first order centered difference in space  
휕푥  
ꢄ푐  
ꢄꢃ  
퐶ꢇꢈ+1 − 퐶ꢇꢈ−1  
2
(
)
≈
+ 푂 ∆ꢃ  
2∆ꢃ  
2
휕 ꢉ  
Discretization of  
is obtain from second order centered difference in space  
2
휕푥  
ꢄ2푐  
ꢄꢃ2  
퐶ꢇꢈ−1 − 2퐶ꢇꢈ + 퐶ꢇꢈ+1  
2
(
)
≈
+ 푂 ∆ꢃ  
∆ꢃ2  
The simplest numerical discretization of (4b) is  
푐ꢇꢈ+1 − 푐ꢇꢈ  
∆푡  
푐ꢇꢈ+1 − 푐ꢇꢈ−1  
푐ꢇꢈ+1 − 2푐ꢇꢈ + 푐ꢇꢈ−1  
+ 푢ꢇꢈ  
= 퐷  
,
2∆ꢃ  
∆ꢃ2  
2
we get,  
푐ꢇꢈ+1 = (ꢌ1ꢍ + ꢋ)푐ꢇꢈ−1 + (1 − ꢌꢍ) 푐ꢇꢈ + (ꢌ1ꢍ − ꢋ2) 푐ꢇꢈ+1  
,
(5b)  
2
∆푡  
퐷∆푡  
∆ꢃ2  
where, 훾 =  
푢ꢇꢈ ,  
= 푝푒 =  
∆ꢃ  
Which is the explicit centered difference scheme and it is also known as FTCS techniques. The stability condition  
is controlled by  
∆ꢁ  
∆ꢁ  
ꢊ∆ꢁ  
훾 =  
푢ꢇꢈ , 푟 = 푝푒 =  
and 훾 = ∆ꢁ 푢ꢇꢈ , = 푝푒 =  
2
2
∆푥  
∆푥  
∆푥  
∆푥  
It is seen that the truncation errors for the forward difference is of first order; whereas the centered difference  
yields a second order truncation error (using by Taylor Series expansions). Therefore, the scheme outlined above  
is consistent.  
Stability analysis of Convection Diffusion equation  
We determine stability conditions of CDE for both the schemes as in the following two propositions.  
Proposition-01  
Statement: The stability conditions of CDE for the FTBSCS scheme are  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and −푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
ꢊ
ꢊ
0 ≤ 푝푒 ≤ 1 and − ∆푥 ≤ max(푢ꢇ0) ≤ ∆푥 − 2 ∆푥  
∆ꢁ  
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Proposition-02  
Statement: The stability conditions of CDE for the FTCS scheme are  
1
∆ꢁ  
and −2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
ꢈ
(
)
0 ≤ 푝푒 ≤  
2
1
2ꢊ  
∆푥  
∆ꢁ  
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and −  
≤ max(푢ꢇ0) ≤ 2 (  
−
∆ꢊ푥).  
2
∆푥  
Numerical Simulation and Results Discussions for CDE  
Various finite difference equations can be used to represent the system of PDE’s which is convection diffusion  
equation (1a), (1b). It is extremely important to experiment with the application of these numerical techniques.  
It is hoped that by writing computer codes and analyzing the results, additional insights into the solution  
procedures are gained. Therefore, this section proposes an example and presents solutions by the described  
schemes.  
Verification of Stability Conditions of CDE  
In this study, we assume that the length of spatial domain, l = 6 meters at all time, t = 1minute to t = 6 minutes  
with viscosity, = 0.01m2/s = 36 m2/h and diffusion coefficient, D = 0.01m2/s = 36 m2/h.  
2
The convection-diffusion equation for this problem is 휕ꢉ + 푢 휕ꢉ = 퐷 휕휕푥ꢉ . Various values of spatial nodes size  
2
휕ꢁ  
휕푥  
and time steps are to be used to investigate the numerical schemes and the effect of steps on stability.  
An attempt is made to solve the stated problem subject to the imposed initial and Neumann boundary conditions  
by the following:  
The FTBSCS and FTCS schemes with  
x = 0.05  
x = 0.05  
x = 0.05  
x = 0.05  
x = 0.05  
t = 0.033 s,  
t = 0.067 s,  
t = 0.100 s,  
nt = 3600, T = 602 sec 푝푒 ≪ 1  
nt = 3600, T = 604 sec 푝푒 ≪ 1  
nt = 3600, T = 606 sec 푝푒 < 1  
t = 0.1192 s, nt = 3600, T = 607.152 sec 푝푒 ≈ 1  
t = 0.122 s, nt = 3600, T = 607.32 sec 푃푒 ≈ 1  
Stability of CDE by FTBSCS and FTCS schemes:  
Case I. When the step sizes are x = 0.05, t = 0.033, 푝푒 ≪ 1  
In this case, both the schemes are to be used as stated previously.  
The stability conditions of FTBSCS is determined by proposition 3.1-01 as  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and −푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
ꢊ
ꢊ
0 ≤ 푝푒 ≤ 1 and − ∆푥 ≤ max(푢ꢇ0) ≤ ∆푥 − 2 ∆푥  
∆ꢁ  
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The stability conditions of FTCS is determined by proposition 3.2-02 as  
1
∆ꢁ  
and −2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
ꢈ
(
)
0 ≤ 푝푒 ≤  
2
1
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and − 2ꢊ ≤ max(푢ꢇ0) ≤ 2 (∆푥  
−
ꢊ ).  
2
∆푥  
∆ꢁ  
∆푥  
ꢊ∆ꢁ  
where, 푝푒 =  
.
2
∆푥  
For this application,  
0
(
)
The value of max 푢ꢇ = 0.02 which satisfies the guaranteed inequality in both the schemes.  
ꢊ∆ꢁ  
0.01×0.033  
훾 = ∆ꢁ max(푢ꢇ0) = 0.033 0.02 = 0.0132 and 푝푒 =  
=
= 0.132  
2
2
( )  
0.05  
∆푥  
0.05  
∆푥  
FTBSCS0 ≤ 0.132 ≤ 1 and −0.132 ≤ 0.0132 ≤ 1 − 20.132 = 0.736  
and  
1
FTCS0 ≤ 0.132  
≤
and −0.264 ≤ 0.0132 ≤ 1.736  
2
Therefore, the stability conditions for both the schemes are satisfied and a stable solution is expected when 푝푒 ≪  
1, convection is dominating the process. The concentration profiles are shown in Figure 4.1.  
Compare the Solutions  
0.02  
FTBSCS  
FTCS  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
0.006  
0.004  
0.002  
0
0
1
2
3
4
5
6
Spatial co-ordinate (x)  
Figure 4.1: Concentration distribution profiles of CDE with x = 0.05, t = 0.033, 푝푒 ≪ 1  
Case II. When the step sizes are x = 0.05, t = 0.067, 푝푒 ≪ 1.  
In this case, both the schemes are to be used as stated previously.  
The stability conditions of FTBSCS is determined by proposition-01 as  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and −푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
ꢊ
ꢊ
0 ≤ 푝푒 ≤ 1 and − ∆푥 ≤ max(푢ꢇ0) ≤ ∆푥 − 2 ∆푥  
∆ꢁ  
The stability conditions of FTCS is determined by proposition-02 as  
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1
∆ꢁ  
ꢈ
(
)
0 ≤ 푝푒 ≤  
and −2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
2
1
2ꢊ  
∆푥  
∆ꢁ  
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and −  
≤ max(푢ꢇ0) ≤ 2 (  
−
∆ꢊ푥).  
2
∆푥  
ꢊ∆ꢁ  
where, 푝푒 =  
.
2
∆푥  
For this application,  
0
(
)
The value of max 푢ꢇ = 0.02 which satisfies the guaranteed inequality in both the schemes.  
ꢊ∆ꢁ  
0.01×0.067  
훾 = ∆ꢁ max(푢ꢇ0) = 0.067 0.02 = 0.0268 and 푝푒 =  
=
= 0.268  
2
2
( )  
0.05  
∆푥  
0.05  
∆푥  
FTBSCS 0 ≤ 0.268 ≤ 1 and −0.268 ≤ 0.0268 ≤ 1 − 20.268 = 0.464  
and  
1
FTCS 0 ≤ 0.268 ≤  
and −0.536 ≤ 0.0268 ≤ 1.464  
2
Therefore, the stability conditions for both the schemes are satisfied and a stable solution is expected. when  
pe≪1, the convection is dominating the process. The concentration profiles are shown in Figure 4.2.  
Compare the Solutions  
0.02  
FTBSCS  
FTCS  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
0.006  
0.004  
0.002  
0
0
1
2
3
4
5
6
Spatial co-ordinate (x)  
Figure 4.2: Concentration distribution profiles of CDE with x = 0.05, t = 0.067 푝푒 ≪ 1  
Case III. When the step sizes are x = 0.05, t = 0.1, 푝푒 ≪ 1.  
In this case, both the schemes are to be used as stated previously.  
The stability conditions of FTBSCS is determined by proposition-01 as  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and −푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
ꢊ
ꢊ
0 ≤ 푝푒 ≤ 1 and − ∆푥 ≤ max(푢ꢇ0) ≤ ∆푥 − 2 ∆푥  
∆ꢁ  
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The stability conditions of FTCS is determined by proposition-02 as  
1
∆ꢁ  
and −2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
ꢈ
(
)
0 ≤ 푝푒 ≤  
2
1
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and − 2ꢊ ≤ max(푢ꢇ0) ≤ 2 (∆푥  
−
ꢊ ).  
2
∆푥  
∆ꢁ  
∆푥  
ꢊ∆ꢁ  
where, 푝푒 =  
.
2
∆푥  
For this application,  
0
(
)
The value of max 푢ꢇ = 0.02 which satisfies the guaranteed inequality in both the schemes.  
∆ꢁ  
0.1  
ꢊ∆ꢁ  
0.01×0.1  
훾 =  
max(푢ꢇ0) =  
0.02 = 0.04 and 푝푒 =  
=
= 0.4  
2
2
( )  
0.05  
∆푥  
0.05  
∆푥  
FTBSCS 0 ≤ 0.4 ≤ 1 and −0.4 ≤ 0.04 ≤ 1 − 20.4 = 0.2  
and  
1
FTCS 0 ≤ 0.4 ≤  
and −0.8 ≤ 0.04 ≤ 1.2  
2
Therefore, the stability conditions for both the schemes are satisfied and a stable solution is expected where  
푝푒 < 1, the process is balanced convection and diffusion. The concentration profiles are shown in Figure 4.3.  
Compare the Solutions  
0.02  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
0.006  
0.004  
FTBSCS  
FTCS  
0.002  
0
0
1
2
3
4
5
6
Spatial co-ordinate (x)  
Figure 4.3: Concentration distribution profiles of CDE with x = 0.05, t = 0.1 푝푒 < 1  
Instability of CDE by FTBSCS and FTCS schemes:  
Case IV. When the step sizes are increased to x = 0.05, t = 0.1192, 푝푒 ≈ 1.  
The stability conditions of FTBSCS is determined by proposition-01 as  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and − 푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
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퐷
∆ꢃ  
∆푡  
퐷
0 ≤ 푝푒 ≤ 1 and −  
≤ max(푢ꢇ0) ≤  
− 2  
∆ꢃ  
∆ꢃ  
The stability conditions of FTCS is determined by proposition-02 as  
1
∆ꢁ  
and − 2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
ꢈ
(
)
0 ≤ 푝푒 ≤  
2
1
2ꢊ  
∆푥  
∆ꢁ  
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and −  
≤ max(푢ꢇ0) ≤ 2 (  
−
∆ꢊ푥).  
2
∆푥  
ꢊ∆ꢁ  
where, 푝푒 =  
.
2
∆푥  
For this application,  
0
(
)
The value of max 푢ꢇ = 0.02 which satisfies the guaranteed inequality in both the schemes.  
ꢊ∆ꢁ  
0.01×0.1192  
훾 = ∆ꢁ max(푢ꢇ0) = 0.1192 0.02 = 0.04768 and 푝푒 =  
=
= 0.4768  
2
2
( )  
0.05  
∆푥  
0.05  
∆푥  
FTBSCS 0 ≤ 0.4768 ≤ 1 and −0.4768 ≤ 0.04768 ≤ 0.0464,  
which does not satisfy the stability condition of FTBSCS scheme,  
and  
1
FTCS 0 ≤ 0.4768 ≤  
and −0.9536 ≤ 0.04768 ≤ 1.0464  
2
In this case, FTBSCS of the CDE shows an instability. The concentration profiles are shown in the following  
Figure 4.4.  
Compare the Solutions  
0.02  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
0.006  
0.004  
FTBSCS  
FTCS  
0.002  
0
0
1
2 3  
Spatial co-ordinate (x)  
4
5
6
Figure 4.4: Concentration distribution profiles of CDE with x = 0.05, t = 0.1192, 푝푒 ≈ 1  
Concentration distribution profiles of CDE with x = 0.05, t = 0.1192 is presented. Here, the stability conditions  
for the scheme FTBSCS of CDE are not satisfied at t = 0.122 whereas the stability conditions for the scheme  
FTCS are satisfied. Since 푝푒 ≈ 1, the transport is spreading out for the FTBSCS scheme.  
Case V. When the step sizes are increased to x = 0.05, t = 0.122, 푃푒 ≈ 1 which is only a fraction of an  
increase over preceding cases.  
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The stability conditions of FTBSCS is determined by proposition-01 as  
∆ꢁ  
0 ≤ 푝푒 ≤ 1 and −푝푒 ≤  
푢ꢇꢈ ≤ 1 − 2푝푒  
∆푥  
This is guaranteed by the simultaneous inequalities  
ꢊ
ꢊ
0 ≤ 푝푒 ≤ 1 and − ∆푥 ≤ max(푢ꢇ0) ≤ ∆푥 − 2 ∆푥  
∆ꢁ  
The stability conditions of FTCS is determined by proposition-02 as  
1
∆ꢁ  
and −2푝푒 ≤ ∆푥 푢ꢇ ≤ 2 1 − 푝푒 .  
ꢈ
(
)
0 ≤ 푝푒 ≤  
2
1
2ꢊ  
∆푥  
∆ꢁ  
This is guaranteed by the conditions 0 ≤ 푝푒 ≤ and −  
≤ max(푢ꢇ0) ≤ 2 (  
−
∆ꢊ푥).  
2
∆푥  
ꢊ∆ꢁ  
where, 푝푒 =  
.
2
∆푥  
For this application,  
0
(
)
The value of max 푢ꢇ = 0.02 which satisfies the guaranteed inequality in both the schemes.  
ꢊ∆ꢁ  
0.01×0.1192  
훾 = ∆ꢁ max(푢ꢇ0) = 0.122 0.02 = 0.0488 and 푝푒 =  
=
= 0.976  
2
2
( )  
0.05  
∆푥  
0.05  
∆푥  
FTBSCS 0 ≤ 0.976 ≤ 1 and −0.976 ≤ 0.0488 ≤ −0.0952,  
which does not satisfy the stability condition of FTBSCS scheme, and  
1
FTCS 0 ≤ 0.976 ≤  
and −1.952 ≤ 0.0488 ≤ 0.024 which does not satisfy the stability condition of  
2
FTCS scheme.  
In this case, both FTBSCS and FTCS schemes of the CDE show an instability. The concentration profiles are  
shown in the following Figure 4.5.  
x 1068  
Compare the Solutions  
2.5  
2
1.5  
1
0.5  
0
-0.5  
-1  
-1.5  
FTBSCS  
FTCS  
-2  
-2.5  
0
1
2 3  
Spatial co-ordinate (x)  
4
5
6
Figure 4.5: Concentration distribution profiles of CDE with x = 0.05, t = 0.122 푝푒 ≈ 1  
Concentration distribution profiles of CDE with x = 0.05, t = 0.122, 푝푒 ≈ 1 is presented. Here, the stability  
conditions for both the schemes FTBSCS and FTCS of CDE are not satisfied at t = 0.122 and 푝푒 ≈ 1 the  
transport is spreading out, like blob.  
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Table 1 displays the numerical solution of our considered problem for different values of diffusion coefficient,  
D = 0.002 m2/s and viscosity coefficient, = 0.002 m2/s for distance, x = 42 meters and time, t = 3000 seconds  
and compare the result with another numerical solution obtained by the spline function and finite elements in [6]  
by using the same values of parameters in the same domain.  
Distance  
x(m)  
D = 0.002 m2/s, = 0.002 m2/s  
Proposed methods for CDE  
Reference method [6]  
SF-FE [74]  
FTBSCS  
0.0200  
FTCS  
0.0  
0.0200  
1.000  
18.0  
19.0  
20.0  
21.0  
22.0  
23.0  
24.0  
25.0  
26.0  
27.0  
28.0  
29.0  
0.0200  
0.0199  
0.0199  
0.0198  
0.0196  
0.0193  
0.0188  
0.0181  
0.0172  
0.0160  
0.0146  
0.0129  
0.0200  
0.0200  
0.0200  
0.0200  
0.0199  
0.0198  
0.0196  
0.0191  
0.0183  
0.0172  
0.0155  
0.0134  
1.000  
0.999  
0.998  
0.996  
0.990  
0.978  
0.957  
0.922  
0.870  
0.799  
0.708  
0.602  
Distance  
x(m)  
D = 0.002 m2/s, = 0.002 m2/s  
Proposed methods for CDE  
Reference method [6]  
SF-FE [74]  
0.488  
FTBSCS  
0.0110  
0.0091  
0.0073  
0.0055  
FTCS  
30.0  
31.0  
32.0  
33.0  
0.0110  
0.0085  
0.0062  
0.0041  
0.375  
0.272  
0.185  
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34.0  
35.0  
36.0  
37.0  
38.0  
39.0  
40.0  
41.0  
42.0  
0.0041  
0.0028  
0.0019  
0.0012  
0.0007  
0.0004  
0.0002  
0.0001  
0.0000  
0.0026  
0.0015  
0.0008  
0.0004  
0.0002  
0.0001  
0.0000  
0.0000  
0.0000  
0.118  
0.070  
0.038  
0.020  
0.009  
0.004  
0.002  
0.001  
0.000  
Table 1: Comparison of results for D = 0.002 m2/s, = 0.002 m2/s with Δt = 6 s and Δx = 0.25 m.  
Selection of Numerical values of parameters for the solution of CDE  
For temporal variable t, we consider the domain 푡(0, 6) in minute and for spatial variable x, we consider the  
domain ꢃ(0, 6) in meter. The initial concentration is considered as 푐0 ꢃ = 푚푎ꢃ푐0 × 푒−10푥, where 푚푎ꢃ푐0 is  
( )  
the maximum of the concentration c(t, x).  
For u = 0.01 m/s and D = 0.001m2/s at time from 1 minute to 6 minutes, the solution of CDE for the numerical  
scheme FTBSCS is shown in Figure 5.1, which shows that the concentration distribution within the described  
domain.  
Solution of CDE  
0.02  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
1 min  
2 min  
0.006  
3 min  
4 min  
0.004  
5 min  
6 min  
0.002  
0
0
1
2
3
4
5
6
Spatial co-ordinate (x)  
Figure 5.1: Solution of CDE by FTBSCS at different time with =0.01 m2/s and D=0.001 m2/s.  
For u = 0.01 m/s = 36 m/h and D = 0.01m2/s = 36 m2/h at time from 1 minute to 6 minutes, the solution of CDE  
for the numerical scheme FTCS is shown in Figure 5.2, which shows that the pollutant distribution within the  
described domain.  
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Solution of CDE  
0.02  
0.018  
0.016  
0.014  
0.012  
0.01  
0.008  
0.006  
0.004  
0.002  
0
1 min  
2 min  
3 min  
4 min  
5 min  
6 min  
0
1
2 3  
Spatial co-ordinate (x)  
4
5
6
Figure 5.2: Solution of CDE by FTCS at different time with =0.01 m2/s and D=0.01 m2/s.  
CONCLUSION  
In this study, we investigated two finite difference schemes—Forward Time Backward Space Centered Space  
(FTBSCS) and Forward Time Centered Space (FTCS)—for solving the convection-diffusion equation (CDE)  
associated with viscous Burgers’ equation. We derived the stability conditions for both schemes and conducted  
a stability analysis to establish constraints on the Peclet number in terms of the time step, spatial step, advection  
coefficient (u), and diffusion coefficient (D).  
Figures 4.1–4.3 demonstrate stable numerical solutions, as the chosen parameters satisfy the derived stability  
conditions. In contrast, Figures 4.4–4.5 exhibit instability due to violations of these conditions within the  
prescribed domain. A comparative analysis with an existing numerical solution from [6], presented in Table 1,  
confirms the accuracy and reliability of our results.  
Finally, in Figures 5.1 and 5.2, we illustrate the numerical solutions of the CDE using artificially selected  
parameter values to further validate the behavior of the schemes. The findings highlight the critical role of  
stability conditions in ensuring accurate and convergent solutions for convection-diffusion problems.  
REFERENCES  
1. Roache, P. J. (1972). Computational Fluid Dynamics. Hermosa Publishers.  
2. Fletcher, C. A. J. (1991). Computational Techniques for Fluid Dynamics. Springer-Verlag.  
3. Berger, M. J. (1984). "Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations." Journal  
of Computational Physics, 53, 484-512.  
4. Changjun Zhu and Shuwen Li, “Numerical Simulation of River Water Pollution Using Grey Differential  
Model,” Journal of computers, Vol. No.9, 2010.  
5. D.J. Evans, A.R. Abdullah, The group explicit method for the solution of Burger’s equation, Computing  
32 (1984) :239-253.  
6. A. Kumar, D. K. Jaiswal and N. Kumar, Analytical solution of one-dimensional Advection diffusion  
equation with variable coefficients in a finite domain, J. Earth Syst. Sci. 118, No.5, pp. 539-549, October  
2009.  
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