ةكلمملا ةيبرعلا
ةيدوعسلا
ةرازو ميلعـــــتلا
يلاـــعلا
ةعماج كلملا
دبع زيزعلا
ةيـــــلك مولـــعلا
تانـــبلل
مـــــــسق تايـــــــضايرلا
ةيمزراوخ قورف
ةرقتسم لحل
ةلداعم قفدتلا
A Stable Difference Algorithm For The Solution of Advection Equation
by
Naimah Ali Al-Malki
Supervisor
Dr. Huda Omar Bakodah
Introduction . 1
Many mathematical models for engineering systems, particularly systems with pronounced flow- through or convective characteristics, involve the
differential group
Which is called the advection group .
However, the simplest partial differential equation containing advection group
1( ) Which is called the advection equation
.
. 2 Finite differences method for solving advection equation
Solve problem with the finite difference method based on three steps :
1 . Dividing the solution area by mesh points .
. 2 Approximate the differential equation given using similar finite differences, which represents the relationship between the values of the solution at the mesh .
3 . Solving the equations of differences with
boundary problems or initial value problems given .
-2 Centered- Differencing Scheme 1
Let us consider the equation
where
and
is a nonzero constant velocity with the initial condition
In the finite difference method we start with discretization the space variable
and the time variable . as follows
and
Using centered - difference scheme yields
Where
To investigate stability of the scheme we follow Von Neumann analysis, we get the growth factor
So the formula is unstable .
.
In our case
for all ,
-2 Upwind Method2
If we use the backward difference yields .
Using Von Neumann analysis the growth factor is
The stability condition is fulfilled for all . as long as
Which is so-called the Courant-Friedrichs-lewy (CFL) condition .
)*(
-2 The Lax Method 3
If we replaced the term by an average
So we obtain the formula .
The stability condition is fulfilled for all
as long as .
Which is again the Courant-Friedrichs-lewy condition .
3
. A good spatial discretization
If we use the new spatial discretization
This yields .
2( )
In order to investigate the stability of this scheme by Von Neumann stability analysis we get
That is, the method ( ) is Stable2
.
.4 Numerical example
.1 Example 1
Consider the nonlinear advection equation
Subject to the following conditions
With the analytic solution
Table 1
shows the absolute error
for the upwind method and good spatial discretization eq(
.)2
Eq (
)2 Upwind Method
0.0206797 0.135379 0.1
0.0105483 0.233035 0.2
0.00921847 0.37713 0.3
0.0085155 0.446792 0.4
0.00790683 0.493141 0.5
0.00863854 0.566328 0.6
0.00712179 0.489712 0.7
0.0611446 0.435176 0.8
0.0798153 0.695396 0.9
0.111210 0.536143 1.1
Table 1
. comparison of the absolute error between upwind method and eq ( 2
), for ex.
1 with
.
.2 Example 2
Consider the equation
Subject to the following conditions
With the analytic solution
Table 2
shows the absolute error for this example .
Eq (
)2 Upwind Method
0.00196377 0.0680513 0.1
0.00392755 0.119927 0.2
0.00589132 0.198615 0.3
0.0078551 0.273427 0.4
0.00981887 0.33333 0.5
0.0128395 0.407076 0.6
0.00119615 0.353221 0.7
0.0860841 0.275832 0.8
0.0960722 0.52337 0.9
0.128795 0.432399 1.0
Table 2
. comparison of the absolute error between upwind method and eq ( 2
),
for ex.
with2
.5 Conclusion
In many cases, advection equation is required to obtain the approximate solutions . For this purpose, we using some finite difference method to
approximate the solution .
The methods introduced in this paper for solving the linear and nonlinear advection equation based on finite difference method .
Deciding on the best choice of the numerical method for a given problem depend on the stability condition .
We present a numerical comparison of the two method, which show that the new discretization introduce more accurate results .
References
1[ ] Z. David and D. paul, Applied Partial Differential Equations, Dover
Publication, Inc., Eash 31
2
nd street, Mineola, N.Y.
11501 , U.S.A.,
. 2002
2[ ] A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory,
Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg, . 2009
3[ ] K. George and E.H. Twizell, Stable Second - Order Finite -difference
methods for linear initial – boundary – value problems, Applied Mathematics Lehers, (19
) (2 ), 2006 146-154
.
4[ ] G.D. Smith Numerical Solution of Partial Differential Equations (Finite
Difference Method), Third Edition, Oxford University Press, . 1985
5[ ] Sharaf, Amr A. and Bakodah, H.O., A good Spatial discretization in the
method of Lines, Applied Mathematics and Computation, Vol.
171-2 , pp.
1253-
, (1263 )2005
.
