COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
A High Order Compact Difference Scheme for Solving the Unsteady
Convection-Diffusion Equation
Z. H. Xie*, J. G. Lin, J. T. Zhou
College of Environmental Science and Engineering, Dalian Maritime University, Dalian, 116026 China
Email: [email protected]
Abstract A high order compact difference scheme based on the fourth order compact difference scheme in spatial
discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of
the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is tested by three
application examples for a two-dimensional convection-diffusion problem with a Gaussian type pulse, a
two-dimensional non-linear Burgers equation and the Taylor’s vortex problem. It is shown that the results obtained by
proposed method agree very well with the analytical solutions and the proposed scheme is not only simple to
implement and economical to use, but also is effective to simulate transport problems.
Key words: convection-diffusion equations, Burgers equation, high order compact difference method, 2D,
Taylor vortex
INTRODUCTION
In practical engineering applications, convection diffusion equations are generally used to describe the transport
processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics,
pollutant & sediment transport and chemical engineering.
Various numerical finite difference schemes have been proposed to solve convection-diffusion problems
approximately. There are two mainly numerical methods for Computational Fluid Dynamics(CFD) to solve the
convection diffusion equations with respect to the temporal discretization, one is the explicit algorithm and the other is
the implicit algorithm. Implicit schemes often exhibit unconditional stability for governing equations, but involve
more complex code, more computational time and storage per iteration. Explicit schemes are usually conditionally
stable, but are relatively easy to program, require less computational time and storage per iteration.
To achieve higher spatial accuracy, recently, the high order compact schemes have been widely utilized for spatial
discretizations. Lele [1] went through an extensive analysis of compact schemes and applied them for solution of
compressible and incompressible flow problems. Afterward, Chu and Fan [2] derived sixth-order and eighth-order
three point combined compact finite difference schemes based on the local and global Hermite polynomials
interpolation. Mahesh [3] extended the derivation of Chu and Fan [2] to more than three point stencil and presented
combined compact uniform grid finite difference schemes which evaluate both the first and the second derivative
simultaneously. Hixon [4] proposed a family of small-stencil compact schemes based on a prefactorization method to
reduce a non-dissipative central-difference stencil to two lower-order biased stencils which have easily solved reduced
matrices. At the same time, Tolstykh and Lipavskii [5], Ma et al [6] developed the third and fifth-order upwind
compact difference schemes. Ma and Fu [7] presented the super compact finite difference method which evaluates all
the derivatives simultaneously. Boersma [8] presented the stagger compact schemes. From what has been discussed
above, it is shown that all the present compact schemes are implicit and consider as unknowns at each discretization
point not only the value of the function but also those of its first or higher derivatives. They will need more
computational costs and are not considered as the exactly compact small-stencil schemes which only need three-point
stencil.
In this paper, a three-point explicit compact difference scheme is proposed. The present scheme employs the
fourth-order accurate approximations for the first and second derivatives in the convection-diffusion equation and
⎯ 231 ⎯
coupling with the fourth-order Runge-Kutta method in time integration. Through the three application examples for a
two-dimensional convection-diffusion problem with a Gaussian type pulse, a two-dimensional non-linear Burgers
equation and the Taylor’s vortex problem, it is found that the proposed scheme is not only simple to implement and
economical to use, but also is effective to simulate transport problems.
GOVERNING EQUATION AND THE NUMERICAL METHOD
1. Governing equation The unsteady convection-diffusion equation for a variable φ in two-dimensional space can be
written as
∂φ
∂φ
∂φ
∂ 2φ
∂ 2φ
= −c x
− cy
+ εx 2 + εy 2 ,
∂t
∂x
∂y
∂x
∂y
φ (0, x, y ) = φ0 ( x, y ),
( x, y ) ∈ Ω
aφ + b
∂φ
= f (t , x, y ),
∂n
( x, y ) ∈ Ω × (0, T ]
(1)
( x, y ) ∈ ∂Ω, t ∈ (0, T ]
where Ω is a rectangular domain, (0,T ] is the time interval, and φ0 and f are the initial and boundary conditions. a
and b are arbitrary coefficient describing the boundary condition as a Dirichlet, Neumann or Robin type in the
boundary normal direction n . cx ( y ) and ε x ( y ) denote the convection velocity and viscosity in the x( y ) -direction,
respectively.
2. Spatial discretization A high order three-point explicit compact difference scheme is used to approximate the
spatial derivatives of the vorticity transport equation. We first introduce the following operators:
φi +1, j − φi −1, j
d xφi , j =
2
, d yφi , j =
φi , j +1 − φi , j −1
2
,
d xxφi , j = φi +1, j + φi −1, j − 2φi , j , d yyφi , j = φi , j +1 + φi , j −1 − 2φi , j ,
where φi , j is the value of the function in the node (i, j ) , which i and j are the index of the x and y direction,
respectively.
Only fourth order of accuracy for the spatial derivative is considered in this paper and the proposed three-point explicit
compact scheme for the first and second derivatives can be obtained as:
∂φi , j
∂x
∂φi , j
∂y
∂ 2φi , j
∂x
2
∂ 2φi , j
∂y
2
=
1
1
(1 − d xx )d xφi , j
Δx
6
+ O ⎡⎣(Δx) 4 ⎤⎦
(2)
=
1
1
(1 − d yy )d yφi , j
Δy
6
+ O ⎡⎣(Δy ) 4 ⎤⎦
(3)
=
1
1
(1 − d xx )d xxφi , j
2
( Δx )
12
+ O ⎡⎣(Δx) 4 ⎤⎦
(4)
=
1
1
(1 − d yy )d yyφi , j
2
( Δy )
12
+ O ⎡⎣(Δy ) 4 ⎤⎦
(5)
where Δx and Δy are the grid spacing along the x and y direction, respectively.
Then, the semi-discretized form of the unsteady convection-diffusion equation can be obtained as
c
ε
⎤
ε
1
1
1
1
∂φ ⎡ cx
= ⎢ − (1 − d xx )d x − y (1 − d yy )d y + x 2 (1 − d xx )d xx + y 2 (1 − d yy ) d yy ⎥ φ
∂t ⎣ Δx
Δy
6
6
(Δx)
12
( Δy )
12
⎦
(6)
3. Time discretization The spatial derivatives were discretized by a high order accurate method, so it should also use
a high-order discretized procedure for the time integration. Hence, the fourth-order accurate Runge-Kutta method is
used in this paper:
⎯ 232 ⎯
1
⎧ *
n
n
⎪φ = φ + 2 ΔtL(φ )
⎪
⎪φ ** = φ n + 1 ΔtL(φ * )
⎪
2
⎨
⎪φ *** = φ n + ΔtL(φ ** )
⎪
1
⎪ n +1
n
n
*
**
***
⎪⎩φ = φ + 6 Δt ⎡⎣ L(φ ) + 2 L(φ ) + 2 L(φ ) + L(φ ) ⎤⎦
(7)
where L(⋅) is the spatial discretized counterparts in Eq. (6).
NUMERICAL EXAMPLES AND RESULTS
1. Convection-diffusion problem with a Gaussian type pulse To examine the validity and effectiveness of the
present method, a widely studied unsteady problem concerning the convection-diffusion of a Gaussian pulse [9-13] is
considered in Eq. (1) with the following initial condition:
⎡ ( x − 0.5) 2
φ (0, x, y ) = exp ⎢ −
⎢⎣
εx
−
( y − 0.5) 2 ⎤
⎥
εy
⎥⎦
(8)
An analytical solution to this problem is
⎡ x − 0.5 − c t 2 ( y − 0.5 − c t )2 ⎤
(
1
y
x )
⎥
exp ⎢ −
φ ( t , x, y ) =
−
⎢
ε x ( 4t + 1)
ε y ( 4t + 1) ⎥
4t + 1
⎣
⎦
(9)
The Dirichlet boundary conditions are taken from the analytical solution.
For the sake of comparison of our results with others, the values of various parameters shown in Table 1 are used
herein.
Table 1 The parameters for the convection-diffusion of a Gaussian pulse
εx
εy
cx
0.01 m2/s 0.01 m2/s 0.8 m/s
cy
Ω
0.8 m/s
0 ≤ x ≤ 2,
0≤ y≤2
Δt
Δx
0.00625 s 0.05 m
Δy
T
0.05 m
1.25 s
The initial pulse and the pulse at t = 1.25 obtained by the present scheme are shown in Fig. 1. Numerical solutions are
compared with the analytical solutions in the diagonal region at the final time step ( t = 1.25 ) in Fig. 2.
1
Φ
0.8
0.6
0.4
0.2
0
0
2
1.5
0.5
1
1
x
0.5
1.5
2
y
0
Figure 1: The initial and the numerical pulse at t = 1.25
⎯ 233 ⎯
0.18
Numerical
Analytical
0.16
0.14
0.12
Φ
0.1
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 2: The comparison of the numerical and analytical results in the diagonal at t = 1.25
Contour plots of the exact and numerically approximated pulses in the sub-region 1 ≤ x, y ≤ 2 are drawn in Fig. 3. Fig.
3(b) show that the present scheme captures very well the moving pulse, yielding pulses centered at (1.5,1.5) and
almost indistinguishable from the exact one displayed in Fig. 3(a).
2
2
1.9
1.9
1.8
1.8
0.
04
0.
1
0. 0
8
1.3
01
0.
01
0.
1.2
0.1
6
1.4
0. 0
8
1.3
1.5
4
0. 1
0.
1
1.4
0.06
y
0.06
y
1.6
16
0.
1.5
4
0. 1
12
0.
1.6
1.7
0.
12
04
0.
1.7
1.2
2
0.0
0.02
1.1
1
1.1
1
1.2
1.4
1.6
1.8
1
2
1
1.2
1.4
1.6
x
x
(a) exact
(b) present scheme
1.8
2
Figure 3: Contour plots of the pulse in the sub-region 1 ≤ x, y ≤ 2 at t = 1.25
The L2 - norm of the errors produced by the present scheme, the P-R ADI scheme [10], the spatial third order
nine-point compact scheme of Noye and Tan [11], the fourth-order nine-point compact scheme of Kalita et al [12], and
the high order ADI scheme of Karaa and Zhang [13], are presented in Table 2. The results show that the present scheme
provides the most accurate solution.
Table 2 L2 -error norms at t = 1.25 delivered by five different schemes,
with Δt = 0.00625 and Δx = Δy = 0.025
L2 -error norms
Method
2.02 ×10−3
1.24 × 10−4
1.02 × 10−4
5.62 × 10−5
2.22 ×10−5
P-R ADI [10]
Noye and Tan [11]
Kalital et al [12]
High order ADI [13]
Present scheme
2. 2D Burgers problem we consider the following solution of a 2D Burgers problem over the square domain
Ω = [0,1] × [0,1] for which an analytical solution can be devised and shown in [14].
⎯ 234 ⎯
The problem is defined by the equations
r
r
r 1 2r
ut + (u ⋅ ∇)u =
∇ u,
Re
(10)
∂
∂
r
j and Re is the Reynolds number and with the following initial and boundary
where u = (u , v) , ∇ = i +
∂x ∂y
conditions:
u ( x, y,0) = sin(π x)cos(π y ), v( x, y,0) = cos(π x)sin(π y ),
u (0, y, t ) = u (1, y, t ) = 0, v( x,0, t ) = v( x,1, t ) = 0,
(11)
∂u
∂u
∂v
∂v
( x,0, t ) = ( x,1, t ) = 0, (0, y, t ) = (1, y, t ) = 0
∂n
∂n
∂n
∂n
Since u and v are similar, only the results of u obtained on a uniform mesh of 30 × 30 with Δt = 1/ 40 from t = 0 to
t = 1 for Re = 100 by the present method are shown in Fig. 4, which agrees well with the results of [14].
t=0.2s
1
1
0.5
0.5
0
U
U
t=0
0
-0.5
-0.5
-1
0
-1
0
0.2
0.2
0
0.4
0.6
y
0.4
y
0.6
0.8
0.2
0.6
0.4
0.6
0.8
0.8
1
0
0.4
0.2
x
1
0.8
1
1
t=0.6s
t=1s
1
1
0.5
0.5
0
0
U
U
x
-0.5
-0.5
-1
0
-1
0
0.2
0.2
0
0.4
0.6
y
0.4
y
0.6
0.8
0.8
1
0
0.4
0.2
0.2
0.6
0.4
0.6
0.8
x
0.8
1
1
x
1
Figure 4: Solution of u ( x, y, t ) of 2D Burgers equation for Reynolds number Re = 100
3. Taylor’s vortex problem Taylor in 1923 published a exact solution of the incompressible Navier-Stokes equations
in terms of the streamfunction and vorticity [15]. In the present paper, the unsteady transport equation of vorticity is
solved by proposed scheme and the Poisson equation of streamfunction is solved by the successive over relaxation
iterative method. The initial conditions is taken as [6,12]
u ( x, y,0) = − cos( Nx)sin( Ny )
and
v( x, y,0) = sin( Nx)cos( Ny )
for
0 ≤ x, y ≤ 2π
(12)
The exact solution of this problem is given by
u ( x, y, t ) = − cos( Nx)sin( Ny )exp −2 N
2
t / Re
and
v( x, y, t ) = sin( Nx)cos( Ny )exp −2 N
2
t / Re
(13)
where N is an integer. Fig. 5 depicts the computed Taylor’s vorticity contours for Δx = Δy = 2π / 64 , Re = 1000 and
N = 4 at t = 2 with Δt = 0.01 . The variations of the horizontal velocity along the vertical centerline and the vertical
⎯ 235 ⎯
velocity along the horizontal centerline at time t = 10 and Re = 100 for N = 1,2,4 , along with the exact solutions are
presented in Fig. 6(a) and (b). It is seen that our results are in good agreement with the exact solutions.
6
5
Y
4
3
2
1
0
0
1
2
3
4
5
6
X
Figure 5: Vorticity contours for the Taylor’s vortex at t = 2 for N = 4 and Re = 1000
1
6
N=1 numerical
N=2 numerical
N=4 numerical
N=1 exact
N=2 exact
N=4 exact
5
N=1 numerical
N=2 numerical
N=4 numerical
N=1 exact
N=2 exact
N=4 exact
0.8
0.6
0.4
4
0
v
y
0.2
3
-0.2
2
-0.4
-0.6
1
-0.8
0
-1
-0.8
-0.6
-0.4
-0.2
0
u
0.2
0.4
0.6
0.8
-1
1
0
1
2
3
4
5
6
x
(a) Horizontal velocity along the vertical centerline
(b) Vertical velocity along the horizontal centerline
Figure 6: Comparison of the computed and the exact velocities for the Taylor’s vortex problem
at t = 10 for N = 1,2, 4 and Re = 100
CONCLUSION
A high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization
and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady
convection-diffusion equation. The validity of the proposed method is firstly tested by a two-dimensional
convection-diffusion equation with a Gaussian pulse type concentration. The L2 error norms are used to measure
differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown
that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than
other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the
proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the
Taylor’s vortex problem is investigated by the proposed method. From the three test problems, it is shown that the
⎯ 236 ⎯
proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems
and also can be applied to many engineering problems.
Acknowledgements
The support of
acknowledged.
the National Natural Science Foundation of China under Grant No. 50479053. is gratefully
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