EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5892
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Qualitative Study on Finite Volume Method for Solving
Linear and Nonlinear Partial Differential Equations

Thekra Abdullah Fayez Alfawaz1, Abdulkafi Mohammed Saeed1,∗

1 Department of Mathematics, College of Science, Qassim University, Buraydah,
Kingdom of Saudi Arabia

Abstract. Numerous numerical solutions have been developed over the past decades to provide
suitable solutions for several types of problems in computational fluid dynamics (CFD). The finite
volume method (FVM) is a numerical technique adopted in computational fluid dynamics to solve
partial differential equations representing conservation laws. In this article, two mathematical
models are presented: one for the linear advection equation and another for the nonlinear Burgers’
equation with diffusion, along with various schemes of the FVM. Furthermore, numerical experi-
ments will be conducted for several FVM schemes to demonstrate the most effective approach for
solving the studied problem.

2020 Mathematics Subject Classifications: 76S05, 76D10, 80A32, 35Q79

Key Words and Phrases: Linear advection equation, nonlinear Burgers’ equation, finite volume
method, computational fluid dynamics, error analysis

1. Introduction

Computational fluid dynamics (CFD) is the analysis of systems involving fluid flow,
heat transfer, and associated phenomena such as chemical reactions using computer-based
simulation. This powerful technique is applied across a broad spectrum of both industrial
and non-industrial fields [1].

The finite volume method (FVM) is a numerical approach that converts partial differ-
ential equations, which describe conservation laws over infinitesimal volumes, into discrete
algebraic equations for finite volumes. It has evolved to address complex situations in com-
putational fluid dynamics and has been adapted to study intricate physical phenomena,
including magneto-elastic semiconductor interactions [2], wave interactions in functionally
graded materials [3], and solitary wave solutions in plasma physics [4].

The FVM process includes a range of computational complexities, from mesh genera-
tion to the application of boundary conditions [1].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5892

Email addresses: 441212381@qu.edu.sa (T. ALfawaz), abdulkafi.ahmed@qu.edu.sa (A.M. Saeed)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 2 of 17

The finite volume method has been used by researchers to solve numerous problems
related to computational fluid dynamics, as in the development of an arbitrary Lagrangian-
Eulerian formulation for compressible flows based on the multi-moment finite volume
method [5]. In addition, FVM is used to describe the Navier-Stokes equations and their
solutions [6, 7] and in incompressible flow problems which cover a wide range of physical
phenomena such as environmental and geophysical flows and the dynamics of mechanical
processes [8, 9].

In the FVM, different schemes exhibit unique characteristics, such as conservativeness,
boundedness, and transportiveness. Examples of these schemes include upwind differ-
encing scheme, central differencing scheme, essentially nonoscillatory (ENO) scheme, and
other related schemes [10, 11].

The main purpose of this paper is to investigate the most efficient schemes in the
finite volume method to solve a one-dimensional linear advection equation and a two-
dimensional nonlinear Burgers’ equation with diffusion. The FVM will be formulated and
applied to solve these problems, and numerical experiments will be performed to compare
the approximate solutions with the exact ones.

The following is the outline of this paper: In Section 2, the central differencing scheme
is introduced, followed by Section 3, which covers the upwind differencing scheme. Section
4 presents a brief description of the Essentially Non-Oscillatory (ENO) scheme, and Section
5 details the Weighted Essentially Non-Oscillatory (WENO) scheme. Section 6 presents
numerical experiments to assess the performance of these schemes. Section 7 discusses the
results. Finally, Section 8 concludes the paper and suggests directions for future research,
particularly focusing on extending another scheme.

2. Central differencing scheme

In this scheme, the computational domain is discretized into a finite number of control
volumes. The main concept is to approximate the fluxes at the boundaries of each control
volume using central differences, which provide second-order accuracy. In particular, the
value of the variable at the cell faces is calculated as the average of the values at the
neighboring cell centers. Mathematically, this is expressed as:

fi+ 1
2
=

fi + fi+1

2
, (1)

where
fi+ 1

2
represents the value at the interface between cells i and i+ 1.

fi and fi+1 represent the values at the centers of cells i and i+ 1, respectively [1].

3. Upwind differencing scheme

This scheme relies on the principle of causality, utilizing information from the direction
of the flow to compute fluxes at the control volume boundaries. Particularly, the value at



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 3 of 17

the cell face is determined using the value from the upstream cell, which helps to maintain
stability and reduce numerical oscillations. Mathematically, this is represented as:

fi+ 1
2
=

{
fi if u > 0,

fi+1 if u < 0,
(2)

where
fi+ 1

2
denotes the flux at the interface between control volumes i and i+ 1.

u is the velocity of the flow.
fi represents the values at the center of cell i [1].

4. Essentially Non-Oscillatory (ENO) scheme

Essentially Non-Oscillatory (ENO) schemes started with the classic paper of Harten,
Engquist, Osher, and Chakravarthy [12]. The ENO schemes are high-order accurate and
effectively capture shocks with sharp, visually monotonic transitions. The schemes of
ENO are well-suited for problems containing both shocks and complicated smooth flow
structures.

Additionally, ENO schemes depending on point values and TVD Runge-Kutta time
discretization, which can reduce computational costs significantly for multi-space dimen-
sions, although ENO schemes choose the smoothest stencil to capture discontinuities from
a set of candidate approximation stencils [13, 14].

5. Weighted Essentially Non-Oscillatory (WENO) scheme

The key advantage of Weighted Essentially Non-Oscillatory (WENO) schemes is their
ability to present high accuracy while avoiding spurious oscillations near discontinuities,
such as shocks or contact discontinuities. Furthermore, WENO schemes are capable of
achieving high-order accuracy (third, fifth, or even higher) in smooth regions, making
them highly attractive for problems requiring high precision. These features make WENO
schemes ideal for simulating fluid dynamics, wave propagation, and other problems involv-
ing discontinuities.

WENO schemes have gained more attention than Essentially Non-Oscillatory (ENO)
schemes, both in algorithm development and applications. Unlike ENO, which is based on
a single interpolation or reconstruction using one selected stencil, WENO employs a convex
combination of interpolations from multiple candidate stencils. The weights in WENO
schemes are specifically designed to retain the ENO property for capturing discontinuities
while restoring the background linear scheme in smooth regions of the solution. This dual
feature ensures accuracy and stability, making WENO schemes particularly effective for
addressing the discontinuities present in many complex problems [13–15].

In the paper, two classifications of the method will be discussed:

(i) The 3rd -order Weighted Essentially Non-Oscillatory (WENO3).

(ii) The 5th -order Weighted Essentially Non-Oscillatory (WENO5).



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 4 of 17

5.1. WENO methodology in FVM framework

A general system of hyperbolic conservation laws:

∂u

∂t
+

d∑
i=1

∂

∂xi
fi(u) = 0, (3)

is considered where d is the space dimension, u is the solution vector, and fi(u) is the
physical flux in the i-th direction.

The computational domain in space is defined as Ω =
[
xL, xR

]
, which is divided

into N cells represented by Ii such as Ii =
[
xi− 1

2
, xi+ 1

2

]
. Following that, the grid size

within the domain is given by ∆xi = xi+ 1
2
− xi− 1

2
, and the cell centers are located at

xi =
1
2

(
xi− 1

2
+ xi+ 1

2

)
where i = 1, 2, . . . , N .

For simplicity, considering the one-dimensional scalar hyperbolic conservation law:

∂u

∂t
+

∂

∂x
f(u) = 0, (4)

with the initial condition: u(x, 0) = u0(x)
Applying the finite volume method to Equation (4) and integrating over the cell Ii =[
xi− 1

2
, xi+ 1

2

]
, the conservation law becomes:

d

dt

∫ x
i+1

2

x
i− 1

2

u(x, t)dx+

[
f(u(x, t))|x

i+1
2

− f(u(x, t))|x
i− 1

2

]
= 0, (5)

Where f(u) is the physical flux, which may be approximated by its numerical counterpart
f̂ (referred to as the numerical flux), and ūi(t) be the average value of the solution u(x, t)
over the cell Ii for all i :

ūi(t) =
1

∆x

∫ x
i+1

2

x
i− 1

2

u(x, t)dx, (6)

where ∆x is the width of the cell.
Substituting this cell-averaged value into the finite volume formulation, the semi-

discrete form of the conservation law becomes:

d

dt
ūi(t) = − 1

∆x

[
f
(
u
(
xi+ 1

2
, t
))

− f
(
u
(
xi− 1

2
, t
))]

, (7)

Where ∆x = xi+ 1
2
− xi− 1

2
approximating Equation (7) using the following conservative

scheme, yields a system of ordinary differential equations (ODEs), leading to the semi-
discrete finite volume scheme:

d

dt
ūi(t) = − 1

∆x

(
f̂i+ 1

2
− f̂i− 1

2

)
, (8)



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 5 of 17

The numerical flux f̂i±1/2 is given by the following expression:

f̂i±1/2 = h

(
u−
i± 1

2

, u+
i± 1

2

)
, (9)

to obtain the semi-discrete finite volume scheme

d

dt
ūi(t) +

1

∆x

[
h

(
u−
i+ 1

2

, u+
i+ 1

2

)
− h

(
u−
i− 1

2

, u+
i− 1

2

)]
= 0, (10)

When the values u±
i+ 1

2

and u±
i− 1

2

were reconstructed byWENOmethod and using the Runge-

Kutta method to solve the resulting of ODEs in (8).
The function h(a, b) in expression (9) is a monotone numerical flux that meets spe-

cific conditions, there are some examples of monotone fluxes include the Godunov flux,
Engquist-Osher flux, and Lax-Friedrichs flux [13, 16].

The goal of WENO reconstruction is to compute the interface values u±
i+ 1

2

and u±
i− 1

2

by

forming a convex combination of all candidate reconstructions u
(r)

i± 1
2

. This involves com-

bining k candidate stencils Sr, each stencil provides sets of neighboring cell values

Sr(i) = {xi−r, xi−r+1, . . . , xi−r+k−1} , r = 0, . . . , k − 1. (11)

For each stencil, generate k different reconstructions for the solution values at the cell
boundaries xi± 1

2
, These reconstructions are computed as:

u
(r)

i+ 1
2

=

k−1∑
j=0

Crj ūi−r+j , u
(r)

i− 1
2

=

k−1∑
j=0

C̃rj ūi−r+j , (12)

where Crj and C̃rj are the reconstructed value at the cell boundary xi+ 1
2
and xi− 1

2
respec-

tively.
The key to the success of WENO would be the choice of the weights ωr, each stencil

is assigned a nonlinear weight ωr, which is computed based on the smoothness indicators
as follows:

ωr =
αr∑k−1
s=0 αs

, ω̃r =
α̃r∑k−1
s=0 α̃s

, (13)

with

αr =
dr

(ε+ βr)
2 , α̃r =

d̃r

(ε+ βr)
2 , r = 0, . . . , k − 1. (14)

where the nonlinear weights satisfy: ωr ≥ 0,
∑k−1

r=0 ωr = 1, and ε is a small number
which must satisfy: ε ≥ 0.

Each stencil is assigned a smoothness indicator βr that is used to measure the smooth-
ness of stencil polynomials, it is used in the calculation of weights, and it constructs weights
such that the discontinuity is avoided inside the stencil, and it’s given as following:

βr =

k−1∑
l=1

∫ x
i+1

2

x
i− 1

2

∆x2l−1

(
∂lpr(x)

∂xl

)2

dx, ∀r = 0, . . . , k − 1. (15)



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 6 of 17

The right-hand side of Equation (15) represents the sum of the squares of scaled L2

norms for all the derivatives of the interpolation polynomial pr(x) evaluated over the
specified interval Ii on the stencil Sr(i). The factor ∆x2l−1 is introduced to remove any
∆x dependency in the derivatives.

If the function u(x) is smooth in all the candidate stencils there are constants dr such
that

ui+ 1
2
=

k−1∑
r=0

dru
(r)

i+ 1
2

= u
(
xi+ 1

2

)
+O

(
∆x2k−1

)
, (16)

ui− 1
2
=

k−1∑
r=0

d̃ru
(r)

i− 1
2

= u
(
xi− 1

2

)
+O

(
∆x2k−1

)
, (17)

where dr is always positive and
∑k−1

r=0 dr = 1, and by symmetry d̃r = dk−1−r. In this
smooth case, to keep the order of accuracy (2k − 1), then ωr should be

ωr = dr +O
(
∆xk−1

)
, r = 0, . . . , k − 1. (18)

which would imply order of accuracy (2k − 1)th order of accuracy

u−
i+ 1

2

=
k−1∑
r=0

ωru
(r)

i+ 1
2

= u
(
xi+ 1

2

)
+O

(
∆x2k−1

)
, (19)

u+
i− 1

2

=
k−1∑
r=0

ω̃ru
(r)

i− 1
2

= u
(
xi− 1

2

)
+O

(
∆x2k−1

)
, (20)

here, ωr and ω̃r, defined in (13), are constructed based on the smoothness indicators
[13, 15–18].

5.2. Time discretization for WENO scheme

After the spatial derivative is discretized with the WENO scheme, a set of ordinary
differential equations (ODEs) is obtained as:

du

dt
= L(u), (21)

where L denotes the spatial discretization operator represented in (8). The generalm-stage
Runge-Kutta (RK) method is written as:

u(i) =

i−1∑
k=0

(
αiku

(k) +∆tβikL
(
u(k)

))
, i = 1, . . . ,m.

u(0) = u(n), u(m) = u(n+1),

(22)



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 7 of 17

The set of ODEs in this paper can be discretized by a 3rd -order by total variation
diminishing (TVD) Runge-Kutta method as follows [13, 19]:

u(1) = u(n) +∆tL
(
u(n)

)
,

u(2) =
3

4
u(n) +

1

4
u(1) +

1

4
∆tL

(
u(1)

)
,

u(n+1) =
1

3
u(n) +

2

3
u(2) +

2

3
∆tL

(
u(2)

)
.

(23)

6. Numerical experiments

In this section, two numerical experiments will be carried out to solve linear advection
of multiple waves and two-dimensional Burgers’ equation with diffusion for comparison
purposes and to verify the results.

6.1. Linear advection of multiple waves

Considering the one-dimensional linear advection equation:

∂u

∂t
+

∂u

∂x
= 0, (24)

With the initial condition:

u(x, 0) =



1
6 [G(x− 1, β, z − θ) +G(x− 1, β, z + θ) + 4G(x− 1, β, z)], if 0.2 ≤ x < 0.4,
1, if 0.6 ≤ x ≤ 0.8,
1− |10(x− 1.1)|, if 1.0 ≤ x ≤ 1.2,
1
6 [F (x− 1, α, a− θ) + F (x− 1, α, a+ θ) + 4F (x− 1, α, a)], if 1.4 ≤ x < 1.6,
0, otherwise,

(25)
where:

G(x, β, z) = e−β(x−z)2 , F (x, α, a) =
√

max (1− α2(x− a)2, 0). (26)

with parameters:

a = 0.5, z = −0.7, θ = 0.005, α = 10, β =
log(2)

36θ2
. (27)

The initial condition consists of a Gaussian pulse, a square wave, a sharp triangle wave,
and a half ellipse arranged from the left to the right in the computational domain x ∈ [0, 2].
The exact solution, representing the theoretical solution for the linear advection equation
with a constant propagation speed c, is:

u(x, t) = u0(x− ct). (28)



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 8 of 17

Figure 1: One-dimensional linear advection of multiple waves in showing the exact solution vs. the numerical
solution, along with the u distributions obtained from the central differencing scheme at simulation time t = 0.5.
The spatial discretization is performed on 200 uniform grid points with a Courant number of CFL = 0.5.

Figure 2: One-dimensional linear advection of multiple waves in showing the exact solution vs. the numerical
solution, along with the u distributions obtained from the upwind differencing scheme at simulation time t = 0.5.
The spatial discretization is performed on 200 uniform grid points with a Courant number of CFL = 0.5.



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 9 of 17

Figure 3: One-dimensional linear advection of multiple waves in showing the exact solution vs. the numerical
solution, along with the u distributions obtained from the WENO3 scheme at simulation time t = 0.5. The
spatial discretization is performed on 200 uniform grid points with a Courant number of CFL = 0.5.

Figure 4: One-dimensional linear advection of multiple waves in showing the exact solution vs. the numerical
solution, along with the u distributions obtained from the WENO5 scheme at simulation time t = 0.5. The
spatial discretization is performed on 200 uniform grid points with a Courant number of CFL = 0.5.



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 10 of 17

Table 1: Error analysis for central differencing scheme in FVM framework at CFL = 0.5.

Nx

scheme
Central differencing scheme

L1 error 0.65607
L2 error 0.5954225
L∞ error 0.94175

L1 error 0.48812
L2 error 0.4425450
L∞ error 0.68826

L1 error 0.4083
L2 error 0.39903100
L∞ error 0.93897

L1 error 0.39833
L2 error 0.43939200
L∞ error 1.1754

Table 2: Error analysis for upwind differencing scheme in FVM framework at CFL = 0.5.

Nx

scheme
Upwind differencing scheme

L1 error 0.50568
L2 error 0.424525
L∞ error 0.60352

L1 error 0.37711
L2 error 0.3384950
L∞ error 0.63623

L1 error 0.27896
L2 error 0.27365100
L∞ error 0.61025

L1 error 0.18559
L2 error 0.20518200
L∞ error 0.47363



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 11 of 17

Table 3: Error analysis for WENO3 and WENO5 in FVM framework at CFL = 0.5.

Nx

scheme
WENO3 WENO5

L1 error 0.47798 0.34751
L2 error 0.41071 0.3282925
L∞ error 0.68641 0.6715

L1 error 0.27409 0.18028
L2 error 0.26617 0.2062550
L∞ error 0.58412 0.45793

L1 error 0.15712 0.089675
L2 error 0.18883 0.13884100
L∞ error 0.51135 0.50284

L1 error 0.077098 0.035365
L2 error 0.11764 0.083419200
L∞ error 0.40532 0.36654

6.2. Burgers’ equation with diffusion

Considering the two-dimensional Burgers’ equation with diffusion:

ut + a

[(
u2

2

)
x

+

(
u2

2

)
y

]
−D (uxx + uyy) = 0, (29)

on the domain [0, 2]2.
when a = 1, the exact solution is provided in Equation (30), and Figure 5 represents this
solution at CFL = 0.5 and the final time t = 0.5.

u(x, y, t) = −2Dπ
cos(π(x+ y)) exp

(
−2Dπ2t

)
2 + sin(π(x+ y)) exp (−2Dπ2t)

. (30)

Figure 5: Exact solution of the two-dimensional Burgers’ equation.



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 12 of 17

Figure 6: Two-dimensional Burgers’ equation with diffusion, displaying the numerical solutions from various
schemes for a diffusion coefficient D = 0.0002 at CFL = 0.5. The spatial discretization of the estimated
solution uest(x, y, t) is performed on a 40× 40 uniform grid. The numerical solutions are shown for the central
differencing scheme (top left), upwind differencing scheme (top right), WENO3 (bottom left), and WENO5
(bottom right).



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 13 of 17

Table 4: Error analysis for central differencing scheme in FVM framework for a diffusion coefficient D = 0.0002
at CFL = 0.5.

Nx ×Ny

scheme
Central differencing scheme

L1 error 8.2763e-07
L2 error 1.093e-0620× 20
L∞ error 2.051e-06

L1 error 4.2254e-07
L2 error 7.8345e-0740× 40
L∞ error 2.0943e-06

L1 error 2.1277e-07
L2 error 5.575e-0780× 80
L∞ error 2.0943e-06

L1 error 1.1087e-07
L2 error 3.9544e-07160× 160
L∞ error 2.0977e-06

Table 5: Error analysis for the upwind differencing scheme in FVM framework for a diffusion coefficient D =
0.0002 at CFL = 0.5.

Nx ×Ny

scheme
Upwind differencing scheme

L1 error 1.3824e-06
L2 error 1.1794e-0620× 20
L∞ error 2.051e-06

L1 error 7.5091e-07
L2 error 8.2257e-0740× 40
L∞ error 2.0943e-06

L1 error 3.9118e-07
L2 error 5.7267e-0780× 80
L∞ error 2.0943e-06

L1 error 1.9917e-07
L2 error 4.0102e-07160× 160
L∞ error 2.0977e-06



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 14 of 17

Table 6: Error analysis for WENO3 and WENO5 in FVM framework for a diffusion coefficient D = 0.0002 at
CFL = 0.5.

Nx ×Ny

scheme
WENO3 WENO5

L1 error 1.6834e-07 5.126e-08
L2 error 2.0541 e-07 6.5106e-0820× 20
L∞ error 3.3108e− 07 1.3088 -07

L1 error 2.2e− 07 5.3135e-08
L2 error 2.5939e-07 6.1183e-0840× 40
L∞ error 4.7755e− 07 9.6984e− 08

L1 error 2.3917 - 07 7.1989e-08
L2 error 2.8305e-07 8.4092e-0880× 80
L∞ error 5.6877e-07 1.6397e-07

L1 error 2.4296 e-07 7.4868e-08
L2 error 2.8789e-07 8.8955e-08160× 160
L∞ error 5.9143e− 07 1.884e− 07

7. Discussion

The linear advection of multiple waves equation displays the effectiveness of different
numerical schemes in capturing the dynamics of multiple waveforms. In the first case,
the initial condition provides a comprehensive test to assess the accuracy of the numerical
methods employed.
Simulations confirm that the central differential scheme despite its simplicity and compu-
tational ease results in relatively high error levels particularly for the L2 and L∞ norms
as presented in Table 1. It is essential to highlight that errors are significantly reduced
when employing the upwind differencing scheme which demonstrates greater stability for
advection-dominated problems as shown in Table 2. This strategy, even so, is still less
accurate compared to the Weighted Essentially Non-Oscillatory (WENO) scheme.
The WENO3 and WENO5 schemes as presented in Table 3 show considerably lower errors
across all norms with WENO5 particularly excelling in its ability to resolve steep gradi-
ents and discontinuities more effectively than lower-order methods. Visual comparisons in
Figure 1, 2, 3 and 4 support these findings.
Figure 7 presents a combined plot that compares the performance of all the schemes, high-
lighting the strengths and weaknesses of each scheme.
They confirm that WENO methods maintain the integrity of the wave shapes over time
while the central differencing scheme and upwind differencing scheme patterns reveal sig-
nificant diffusion and distortion of the wave profiles.



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 15 of 17

Figure 7: Linear advection of multiple waves using various schemes at simulation time t = 0.5. The spatial
discretization is performed on 200 uniform grid points with a Courant number of CFL = 0.5.

The two-dimensional Burgers’ equation with diffusion and the numerical solutions
received from different schemes provide insights into the interplay between advection and
diffusion.
The exact solution for the diffusion coefficientD = 0.0002 as presented in Figure 5 provides
a benchmark for estimating the performance of numerical methods.
The error analysis presented in Tables 4 and 5 implies that the central differencing scheme
achieves an acceptable level of accuracy but the errors remain substantial, particularly in
the L∞ norm while the upwind differencing scheme improves upon this improvement but
remains incapable of capturing the finer details of the solution.
The WENO3 and WENO5 schemes in Table 6, continue to excel the lower-order methods
and achieve substantially lower error values.
This is particularly apparent in their ability to resolve the solution’s features as represented
in Figure 6. The high fidelity of the WENO schemes especially is essential for applications
involving sharp gradients where diffusion plays a critical role in the solution behavior.

8. Conclusion

This study rigorously examines numerical solutions to the one-dimensional linear ad-
vection equation and the two-dimensional Burgers’ equation using the Weighted Essen-
tially Non-Oscillatory (WENO5) method. The results demonstrate significantly lower
error rates across various norms, including L1, L2, and L∞. The superior performance of
WENO5 is particularly evident when compared to lower-order schemes.
Specifically, this research establishes that WENO5 outperforms other methods, such as



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 16 of 17

WENO3, central differencing, and upwind differencing, particularly in scenarios charac-
terized by sharp gradients. This capability highlights WENO5’s proficiency in addressing
complex fluid dynamics problems where accuracy is paramount.

Future research may focus on extending other advanced numerical schemes, such as the
Targeted Essentially Non-Oscillatory (TENO) scheme. Investigating its methodological
techniques could lead to further improvements in computational efficiency and accuracy,
potentially broadening its applicability to a diverse range of fluid dynamics challenges.

Acknowledgements

The Researchers would like to thank the Deanship of Graduate Studies and Scientific
Research at Qassim University for financial support (QU-APC-2025). They also extend
their appreciation to the anonymous reviewers for their insightful suggestions, which have
greatly improved the quality of the manuscript.

References

[1] Fadl Moukalled, Luca Mangani, Marwan Darwish, F Moukalled, L Mangani, and
M Darwish. The finite volume method. Springer, 2016.

[2] Kh Lotfy, AA El-Bary, W Hassan, and MH Ahmed. Hall current influence of
microtemperature magneto-elastic semiconductor material. Superlattices and Mi-
crostructures, 139:106428, 2020.

[3] Kh Lotfy and RS Tantawi. Photo-thermal-elastic interaction in a functionally graded
material (fgm) and magnetic field. Silicon, 12(2):295–303, 2020.

[4] Abdulghani R Alharbi, MB Almatrafi, and Kh Lotfy. Constructions of solitary trav-
elling wave solutions for ito integro-differential equation arising in plasma physics.
Results in Physics, 19:103533, 2020.

[5] Peng Jin, Xi Deng, and Feng Xiao. An ale formulation for compressible flows based
on multi-moment finite volume method. Engineering Applications of Computational
Fluid Mechanics, 12(1):791–809, 2018.

[6] Stelian Ion and Anca Veronica Ion. A finite volume method for solving generalized
navier-stokes equations. Annals of the Academy of Romanian Scientists: Series on
Mathematics and its Applications, 3:145–163, 2011.

[7] David Trebotich and Daniel Graves. An adaptive finite volume method for the incom-
pressible navier–stokes equations in complex geometries. Communications in Applied
Mathematics and Computational Science, 10(1):43–82, 2015.

[8] Walter Boscheri, Andrea Chiozzi, Michele Giuliano Carlino, and Giulia Bertaglia. A
new family of semi-implicit finite volume/virtual element methods for incompress-
ible flows on unstructured meshes. Computer Methods in Applied Mechanics and
Engineering, 414:116140, 2023.

[9] Dartzi Pan. A high-order finite volume method solving viscous incompressible flows
using general pressure equation. Numerical Heat Transfer, Part B: Fundamentals,
82(5):146–163, 2022.



T. ALfawaz, A.M. Saeed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5892 17 of 17

[10] Rainer Ansorge and Thomas Sonar. Mathematical models of fluid dynamics: Mod-
elling, theory, basic numerical facts-An introduction. John Wiley & Sons, 2009.

[11] Bram van Leer and Kenneth G Powell. Introduction to computational fluid dynamics.
Encyclopedia of Aerospace Engineering, 5:12–16, 2010.

[12] Ami Harten, Bjorn Engquist, Stanley Osher, and Sukumar R Chakravarthy. Uni-
formly high order accurate essentially non-oscillatory schemes, iii. Journal of compu-
tational physics, 131(1):3–47, 1997.

[13] Chi-Wang Shu. Essentially non-oscillatory and weighted essentially non-oscillatory
schemes. Acta Numerica, 29:701–762, 2020.

[14] Shuhai Zhang, Jun Zhu, and Chi-Wang Shu. A brief review on the convergence to
steady state solutions of euler equations with high-order weno schemes. Advances in
Aerodynamics, 1:1–25, 2019.

[15] Pawel Buchmüller. High-Order WENO Finite Volume Methods on Cartesian Grids
with Adaptive Mesh Refinement. PhD thesis, Dissertation, Düsseldorf, Heinrich-
Heine-Universität, 2016, 2016.

[16] Y-T Zhang and C-W Shu. Eno and weno schemes. In Handbook of numerical analysis,
volume 17, pages 103–122. Elsevier, 2016.

[17] Mohammad Afzal Shadab, Dinshaw Balsara, Wei Shyy, and Kun Xu. Fifth order
finite volume weno in general orthogonally-curvilinear coordinates. Computers &
Fluids, 190:398–424, 2019.

[18] Shantanu Shahane, Sheide Chammas, Deniz A Bezgin, Aaron B Buhendwa, Steffen J
Schmidt, Nikolaus A Adams, Spencer H Bryngelson, Yi-Fan Chen, Qing Wang, Fei
Sha, et al. Rational-weno: A lightweight, physically-consistent three-point weighted
essentially non-oscillatory scheme. arXiv preprint arXiv:2409.09217, 2024.

[19] Fei Teng, Li Yuan, and Tao Tang. A speed-up strategy for finite volume weno schemes
for hyperbolic conservation laws. Journal of Scientific Computing, 46:359–378, 2011.