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KSCE Journal of Civil Engineering (0000) 00(0):1-6
Copyright ⓒ2014 Korean Society of Civil Engineers
DOI
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pISSN 1226-7988, eISSN 1976-3808
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Geotechnical Engineerin g
Extended Cubic B-spline Solution of the Advection-Diffusion Equation
Dursun Irk*, dris Da **, and Mustafa Tombul***
Received December 18, 2013/Revised May 5, 2014/Accepted May 18, 2014/Published Online
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Abstract
The collocation method based on the extended B-spline functions as trial functions is set up to find numerical solutions of the
advection-diffusion equation numerically. The transport of the pollution is simulated by way of using solutions of the advection-
diffusion equation. Comparative results of use of both B-spline and extended B-spline in the collocation method are illustrated. It is
concluded that a suitable selection of the free parameter of extended B-spline is shown to provide less error for the numerical
solutions of the advection-diffusion equation with some boundary and initial conditions.
Keywords: B-spline, collocation method, pollutant transport
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1. Introduction
Since pollutants can be negative effects on the environment,
prediction of spreading of pollutant in streams is very important
and also accurate estimation of the transportation of pollutants is
very significant for effective water quality management. The
basic advection-diffusion equation for contaminant transport is a
standard mathematical model. So solutions of the advection-
diffusion equation have been dealt with by many scientists to
illuminate the pollutant phenomena. Since both advection and
diffusion terms exit in the advection-diffusion equation, it also
arises very frequently in transferring mass, heat, energy and
vorticity in engineering and chemistry. Thus the advection
diffusion equation has also been used to model both physical
and chemical phenomena such as heat transfer in a draining
film, dispersion of tracers in porous media, the intrusion of salt
water into fresh water aquifers, the spread of pollutants in
rivers and streams, the dispersion of dissolved material in
estuaries and coastal sea contaminant dispersion in shallow
lakes contaminant dispersion in shallow lakes, the absorption
of chemicals into beds, the spread of solute in a liquid flowing
through a tube, long-range transport of pollutants in the
atmosphere forced cooling by fluids of solid material such as
windings in turbo generators, thermal pollution in river
systems, flow in porous media and some economics and
financial forecasting (Korkmaz and Dag, 2012).
When initial and boundary conditions are complex and the
advection term is dominant, finding analytical solutions is not
easy task for the equation. Not to obtain unreal solutions of the
advection-diffusion equation with relatively high advection
constant, effective algorithms need to be constructed. Thus a
variety of numerical methods has been developed to solve the
advection-diffusion equation. The numerical methods are
constructed by using spline functions to get the solutions of
differential equations. In the literature, one-dimensional
advection-diffusion equation has been solved with various
methods accompanied with spline functions by many
researcher. Pepper et al. (1979) and Okamoto et al. (1998) have
solved the one-dimensional advection equation by using a
spline interpolation technique called a quasi-Lagrangian cubic
spline method. The characteristic methods integrated with
splines have been proposed to solve advection-diffusion
equation (Szymkiewicz, 1993; Tsai et al., 2004; 2006). Gardner
and Dag (1994) have set up the cubic B-spline Galerkin
method to solve the advection-diffusion equation. The
advection-diffusion equation have been solved numerically by
a quadratic B-spline subdomain collocation method by Gardner
et al. (1994). Funaro and Pontrelli (1999) have solved the
advection-diffusion equation with spline approximation with
the help of upwind collocation nodes. Exponential spline
interpolation in characteristic based scheme for solving the
advective--diffusion equation has been presented in the study
(Zoppou et al., 2000). Ahmad (2000) and Ahmad & Kothyari
(2001) have solved the one dimensional advection-diffusion
equation by using cubic spline interpolation for the advection
component and the Crank-Nicolson scheme for the diffusion
component. A meshless method based on thin-plate spline
radial basis functions has been presented by Boztosun et al.
(2002). Least-squares B-spline finite element method has been
constructed in study of Dag et al. (2006). A comparison of two
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TECHNICAL NOT E
*Associate Professor, Eski ehir Osmangazi University, Dept. of Computer-Mathematics 26480, Turkey (E-mail: dirk@ogu.edu.tr)
**Professor, Eski ehir Osmangazi University, Dept. of Computer-Mathematics 26480, Turkey (Corresponding Author, E-mail: idag@ogu.edu.tr)
***Professor, Anadolu University, Civil Engineering Dept., Eski ehir, Turkey (E-mail: mtombul@anadolu.edu.tr)
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Dursun Irk, dris Da , and Mustafa TombulI
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types of spline solutions methods has been done with standard
finite difference method in the studies of Thongmoon (2006;
2008). A cubic B-spline collocation method has been introduced
by Goh et al. (2010; 2012). Quadratic/Cubic B-spline Taylor-
Galerkin method for advection-diffusion equation have been
proposed in the study (Dag et al., 2011). Kapoor and Dhawan
(2010) and Dhawan et al. (2011; 2012) have proposed cubic/
quadratic B-spline least-squares finite element techniques for
advection-diffusion equation. Numerical solutions of advection-
diffusion equation have been given in the work of Korkmaz and
Dag (2012) by way of constructing cubic B-spline differential
quadrature method. The solution of advection diffusion equation by
the quadratic Galerkin finite elements method has been given in
the work (Bulut et al., 2013).
The several extensions of the cubic B-splines have been
proposed in the studies (Han and Liu, 2003; Wang and Wang
2004; Wang and Wang, 2005; Xu and Wang, 2008). The B-
spline functions are generalized by adding higher order terms to
the piece-wise parts of the B-spline functions having one free
parameter into the extended B-spline functions by keeping the
continuity. The shape of the extended B-spline can be changed
by using the different free parameters. The effect of the
additional term and free parameters for the extended B-splines
have been shown by solving some standard boundary value
problems (Hamid et al., 2010; Hamid et al., 2011; Goh et al.,
2011). Extended B-spline functions are not as widespread as the
B-spline functions to form the numerical methods for finding the
numerical solutions of the partial differential equations. Recently
the method of the collocation based on the extended cubic B-
spline has been described for solving one-dimensional heat
equation with a nonlocal initial condition in the study (Goh et al.,
2011). The study of Dag and his coworkers has appeared to solve
the modified regularized long-wave equation using the method
of the collocation accompanied the extended B-spline functions
(Dag et al., 2013).
The advection-diffusion equation governing the flow along a
uniform straight channel
(1)
will be considered over the region together boundary
conditions
(2)
and the initial condition , where is a
function of two independent variables t and x, which generally
denote time and space, respectively. In the one dimensional linear
advection diffusion equation, α is the steady uniform fluid velocity,
µ is the constant diffusion coefficient and φ in the boundary
condition (2) is the flux at the downstream boundary x = b.
In this paper, the extended cubic B-spline functions are used to
set up method of collocation to get the numerical solutions of the
advection-diffusion equation. Free parameter of the extended B-
splines is sought over the predetermined interval to obtain the
best numerical solution for the advection-diffusion equation.
Numerical solutions are compared with both analytical results
and that of the B-spline collocation methods.
Extended Cubic B-spline Collocation Method
Problem domain of [a, b] is divided into sub-elements at the
knots
where . On this partition,
the extended cubic B-splines , have the
following form:
(3)
Note that the spline Qm(x) and its fourth principle derivatives
does not vanish outside the interval and it degenerates into the
cubic B-spline when . Analogous to the cubic B-splines
(Prenter, 1989), the set of the extended cubic B-splines
form a basis over .
Over the problem domain, the approximate solution
to the exact solution can be written as a combination of
the extended cubic B-splines
(4)
where are time dependent unknown parameters which will be
determined from the collocation method and the boundary and
initial conditions.
The values of the extended cubic B-spline and its
successive derivatives at the knots are listed in the
Table 1 and dashes denote the differentiation with respect to
space variable x.
Using Table 1, approximation UN and its first and second space
derivatives at the knots can be computed as
UtαUxµUxx
–+0=
axb≤≤
Uat,()U0µ∂Ubt,()
∂x
-------------------,φt()t0T,](∈,–==
Ux0,()fx()=UUxt,()=
ax
0x1…xN
<< < b==
hx
mxm1–
–ba–()Nm,⁄ 1…N,,== =
Qmm,1…N1+,,–=
Qmx() 1
24h4
-----------
4h1λ–()xx
m2–
–()
33λxx
m2–
–()
4
+ xm2– xm1–
,[]
4λ–()h412h3xx
m1–
–()6h22λ+()xx
m1–
–()
2
++
12hx x
m1–
–()
3
–3λxx
m1–
–()
4
– xm1– xm
,[]
4λ–()h412h3xm1+ x–()6h22λ+()xm1+ x–()
2
++
12hx
m1+ x–()
3
–3λxm1+ x–()
4
–xmxm1+
,[]
4h1λ–()xm2+ x–()
33λxm2+ x–()
4
+ xm1+ xm1+
,[]
0 otherwise
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎧
=
λ0=
Q1– Q0…QNQN1+
,,,,{} ab,[]
UNxt,()
Uxt,()
UNxt,() δjQj
j1–=
N1+
∑
=
δj
Qmx()
Q′x() Q″x(),
Table 1. Values of Qm(x) and Its Principle Two Derivatives at the
Knot Points
xx
m
Qm00
000
00
xm2– xm1– xm1+ xm2+
4λ–
24
-----------8λ+
12
-----------4λ–
24
-----------
Q′
m
1
2h
------1
2h
------
–
Q″
m
2λ+
2h2
-----------2λ+
h2
-----------
–2λ+
2h2
-----------

Extended Cubic B-spline Solution of the Advection-Di_usion Equation
Vol. 00, No. 0 / 000 0000 −3−
(5)
Returning to the advection-diffusion Eq. (1), time integration
of the equation using the Crank-Nicholson rule yields semi-
integrated advection-diffusion equation:
(6)
Substituting Eqs. (5) into Eq. (6) at the knots
leads to the system of equations:
(7)
where
The system (7) consists of N + 3 equations in N + 1 unknowns.
One eliminates the parameters from the system (7)
using the boundary conditions and requiring that
To carry on the iteration of the system (7), the initial parameters
must be obtained from the initial
condition and the derivatives of the boundary conditions at both ends:
This yields a matrix system where the solution
can be found by Thomas algorithms. Nodal values at the
knots can determined in terms of element
parameters having been found from the system (7) at time
levels n∆t.
2. Numerical Tests
2.1 Problem 1 : Pure Advection in a Infinitely Long Channel
Advection-diffusion equation with boundary condition with µ = 0
and initial conditions
has the exact solution
with standard deviation ρ = 264 m and x0 = 2 km.
During the program run, this initial condition is propagated in
a long channel without change in shape or size by the time t =
9600s using the flow velocity α = 0.5 m/s. So initial condition
travels from the initial position to a distance of 4.8 km and the
peak value of the solution remain constant 10 for all time.
To be able to apply the collocation method, the problem domain
is discretized into elements of length h= 100 m. The program
was run with time step ∆t= 50 and to find best parameter λ for
extended cubic B-spline function, region was scanned
with the increment = 0.01. Error norm against λ is drawn at
time t = 9600s in Fig. 1 to show the least error for the best value
of the parameter λ. Wave shapes are visualized at various times
in Fig. 2. with λ= 0. Variation of absolute error versus spatial
variables are depicted to show the error with parameters λ = 0,
−0.176 in Figs. 3-4.
The use of the extended B-splines in the collocation method
provides less error than using the B-spline in the method. Error
norms for various parameters are also given in the Table 2 from
which the accuracy of the suggested method increases to be nearly
twice better than results of the B-spline collocation method.
2.2 Problem 2: Convection and Diffusion with Continuous
Inflow of Pollutant
In this part of the paper transportation of pollutant with
UmUx
m
() 4λ–
24
-----------δm1–
8λ+
12
----------- δm
4λ–
24
-----------δm1+
++==
U′mU′xm
() δm1–
2h
----------δm1+
2h
----------
+–==
U″mU″xm
() 2λ–
2h2
-----------δm1–
2λ+
h2
----------- δm
–2λ+
2h2
----------- δm1+
+==
Un1+ Un
–t∆
2
-----αUx
n1+ Ux
n
+()µUxx
n1+ Uxx
n
+()–[]+0=
xmm,0…N,,=
α1αt∆
2
-----γ1µt∆
2
-----β1
–+ δm1–
n1+ α2µt∆
2
-----β2
–δm
n1+ ++
α1αt∆
2
-----γ2
+µt∆
2
-----β1
–δm1+
n1+ α1αt∆
2
-----γ1
–µt∆
2
-----β1
+δm1–
n+=
α2µt∆
2
-----β2
+δm
nα1αt∆
2
-----γ2
–µt∆
2
-----β1
+δm1+
n
+
α1
4λ–
24
----------- α2
,8λ+
12
----------- γ1
,1
2h
------ γ2
,–1
2h
------β1
,2λ+
2h2
-----------β2
,2λ+
h2
-----------
–======
δ1–
n1+ δN1+
n1+
,
Ux
0t,() Ux
Nt,()
U0Ux
0
() 4λ–
24
-----------δ1–
8λ–
12
-----------δ0
4λ–
24
-----------δ1
++==
U′
NU′xN
() δN1–
2h
----------
–δN1+
2h
----------
+==
δ0δ1–
0δ0
0…δN
0δN1+
0
,,, ,()=
UmUx
m
() 4λ–
24
-----------δm1–
8λ+
12
-----------δm
4λ–
24
-----------δm1+ m,++ 0…N,,== =
U′0U′x0
() δ1–
2h
------
–δ1
2h
------ U′N
,+U′xN
() δN1–
2h
----------
–δN1+
2h
----------
+== = =
N1+()N1+()×
Um
n1+
xmm,0…N,,=
δm
n
U0t,()0∂U9000 t,()
∂t
----------------------------,0t0>,==
Ux0,()10exp x2000–()
2
2ρ
-------------------------
–
⎝⎠
⎛⎞
=
Uxt,()10exp x2000–αt–()
2
2ρ2
------------------------------------
–
⎝⎠
⎛⎞
=
0.5–0.5,[]
L∞
L∞
Fig. 1. L error Norm with -0.5 ≤ λ ≤ 0.5

Dursun Irk, dris Da , and Mustafa TombulI
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convection and advection will be simulated in a long channel.
Pollutant at the left boundary is kept fixed at the concentration of
and at far right problem domain
is taken for t > 0. The initial condition is for ,
The exact solution of the advection-diffusion is
where erfc(x) is complementary error function. A flow with
velocity α = 0.01 and diffusion coefficient µ = 0.002 are taken
and the channel length [0,200] is divided into uniformly sized
elements of length h = 1 m. The maximum error at time t =
3000s is given in the Table 3 for different time steps with fixed
space step h = 1 m. The results found with the extended B-spline
collocation method is the almost same with that of the B-spline
collocation method. Graphical solutions are shown at times t =
3000, 6000s in Fig. 5.
3. Conclusions
Solutions of the advection-diffusion are investigated using
extended cubic B-spline finite element method. Numerical examples
are presented to demonstrate the efficiency of the method. The
comparison of results obtained with presented method shows that
the improvement of accuracy has been observed for transportation
of the pollutant with appropriate choice of free parameter of the
extended cubic B-splines. Unfortunately, when the solution of
the advection diffusion equation has the type of the sharp behavior as
seen in problem 2, we observe not much improvement w hen the
free parameter of extended cubic B-spline varies in the defined
interval. The B-spline solutions of the advection-diffusion can be
U0t,()1=
∂U200 t,()
∂x
-------------------------0=
Ux0,()0= x0≥
Uxt,()1
2
---erfc xαt–
4tµ
-------------
⎝⎠
⎛⎞
1
2
---exp αx
µ
------
⎝⎠
⎛⎞
erfc xαt+
4tµ
-------------
⎝⎠
⎛⎞
+=
Fig. 2. Wave Profiles
Fig. 3. Absolute Error with λ = 0
Fig. 4. Absolute Error with λ = −0.176
Table 2. L∞ Error for Various Space and Time Steps
h∆tL
∞L∞
200 50 1.29 6.07 × 10−1 (λ = −0.2790)
100 50 3.25 × 10−1 (λ = 0) 5.05 × 10−2 (λ = −0.17600)
50 50 1.98 × 10−1 (λ = 0) 2.20 × 10−3 (λ = −0.50900)
10 10 7.51 × 10−3 (λ = 0) 3.44 × 10−6 (λ = −0.50036)
1 1 7.50 × 10−5 (λ = 0) 3.49 × 10−10 (λ = −0.50004)
0.5 0.5 1.88 × 10−5 (λ = 0) 5.50 × 10−11 (λ = −0.50000)
Table 3. L∞ Error at Some Different Time Steps with h = 1 m
∆tL
∞L∞
60 4.33 × 10−2 (λ = 0) 4.25 × 10−3 (λ = 0.083)
30 1.962 × 10−2 (λ = 0) 1.961 × 10−2 (λ = −0.0672)
20 1.27 × 10−2 (λ = 0) 1.26 × 10−2 (λ = −0.0619)
10 6.85 × 10−3 (λ = 0) 6.08 × 10−3 (λ = −0.0557)
5 4.09 × 10−3 (λ = 0) 3.07 × 10−3 (λ = −0.044)
1 2.24 × 10−3 (λ = 0) 1.27 × 10−3 (λ = −0.02)

Extended Cubic B-spline Solution of the Advection-Di_usion Equation
Vol. 00, No. 0 / 000 0000 −5−
found in the studies documented in section of bibliography so
that comparison of the presented results found with that of some
other numerical methods can be done. We have not compared
our results with those of some other numerical methods because
the results of the B-spline collocation method has already
compared with that of methods given in the studies (Korkmaz
and Dag, 2012; Goh et al., 2012; Dag et al., 2006; Kapoor and
Dhawan, 2010; Dhawan et al., 2012; Gardner et al., 1994; Dag et
al., 2011) Thus the extended B-splines together with collocation
method provides reliable results to find numerical solutions of
the time-dependent partial differential equations.
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Fig. 5. Solutions for Method with λ = 0 and ∆t = 60 s

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