149603-1818-ijcee-ijens

International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 03 17

149603-1818-IJCEE-IJENS June 2014 IJENS I J E N S

Numerical Solution for Diffusion Waves equation

using Coupled Finite Difference and Differential

Quadrature Methods ABDULRAZAK H. AL-MALIKI M. EZZELDIN , & A. S. AL-GHAMDI

PhD student, Department of Civil Engineering, College of Engineering , King Abdulaziz University (KAU),Jeddah ,Saudi Arabia Professor, Department of Civil Engineering, College of Engineering , King Abdulaziz University (KAU),Jeddah ,Saudi Arabia

Abstract-One of the simple and most practical equations that

is used in hydrologic and hydraulic routing, is the Diffusion

Wave equation. Considering the fact that this equation has

an analytical solution only in a specific condition, using

numerical methods for solving it has been common and

finding a good numerical method for solving this equation,

has been the focus of many researchers. The differential

quadrature method (DQM) is one of the numerical methods

that because of its stability . Finite difference method (FDM)

is the most practical method that is used in solving partial

differential equations . It is shown that use of the (DQM),

with (FDM), yields a good convergence of results. The

results have been compared with numerical schemes

available in literature and it shows good agreement with

them. In this research, we have used the features of both

methods for solving Diffusion wave equations. This

research shows that the DQ method is not very sensitive

when it comes to choosing a test function. But when it comes

to distribution of grid points, the cosine distribution gives a

much better result than a uniform distribution. In general

the coupled DQM and FDM method gives an accurate

result in solving the diffusion wave equation, even with few

grid points and the numerical model prepared is very stable.

Results shows that the normal error are 1.3084 , 1.259489

and 0. 96009498 for McCormack , DQ and coupled (DQ)

with (FD) methods , respectively , it is clear that using

coupled (DQ) with (FD) method is providing 21.7 %

decreased in normal error .

Index Term-- Flow routing, Diffusion wave equation, Numerical Solution, Differential Quadrature

1- INTRODUCTION Problems regarding flow routing in an open channel in one

dimension situations, normally analyzed using the saint venant

equations. Solving these problems require complete

information about initial and boundary conditions of the flow.

Also because these equations are nonlinear, in some cases,

especially when there is a sudden change in the angle of the

slope or the cross section, stability problems can arise. That is

why efforts has been made to simplify these equations, two of

which are the general wave kinematics model and wave

diffusion model. Using each of these models depends on the

importance of the effect of pressure gradient term, local

acceleration and convective acceleration in the momentum

equation. In a way that to get to the diffusion wave equation,

the effect of inertial force (local acceleration and convective

acceleration) is ignored and in the kinematics wave equation,

inertia force and also the pressure gradient term are ignored.

These equations have analytical solution in specific condition

such as using channel with simple geometry or in constant

rainfall intensity. Yet, since an analytical solution is not

possible for all problems, numerical methods are used to solve

these problems. The traditional methods of numerical schemes

can be divided into three categories: Finite Difference Method

(FDM), Finite Elements Method (FEM) and Finite Volume

Method (FVM). In some of the previous researches, to solve

the diffusion Wave equations, the finite difference method and the finite element method has been used (Lal, A.M et,

2012, Tommaso et al, 2012, Moussa and Bocquillon et, 2000).

In each of these methods, in order to reach an accurate result,

many grid points have been used.

Differential Quadrature (DQ) is also one of the numerical

methods that is known as a highly applicable method in a lot

of scientific fields. This method was first introduced by

Belman et in 1972. After that Differential Quadrature

methods based on polynomial expansion (PDQ) and

Differential Quadrature based on the Fourier series (FDQ)

were introduced (shu & ching, 1997). Thus, a great progress

was made in using the Differential Quadrature method and it

was used in solving structural analysis problems, flow and

also free vibration of plates problems. (Chen, 2000, Shu et,

2004). What caused the spread of this method, was the use of

less grid points in calculations while maintaining the stability

without any conditions and accuracy of results.

In many of the previous researches, the DQ or FD method

were used alone in solving partial differential equation (PDE),

in this regard, to maximize the efficiency of solving (PDE) ,

DQ and FD method are coupled in calculating spatial

derivatives and estimating time derivatives. The calculation

field is divided into several sections in the direction of time

and the coupled DQ and FD method is used in each section.



In this research, that has been done with the goal of measuring

the performance and efficiency of the coupled DQ and FD

method in solving diffusion wave problems, using the coupled

DQ and FD method, a numerical model with high efficiency

and accuracy has been presented. To assess the DQ - FD

method, the results of these methods have been compared with

numerical schemes available in literature and it shows good

agreement with them .

2- DIFFUSION WAVE PROBLEMS The mathematical representation of the unsteady flow is

governed by the hyperbolic fully non-linear SaintVenant equations, which are difficult to solve analytically. However,

from the highly non-linear dynamic equations, simplified

models such as the kinematic-wave and diffusion-wave model

can be derived. Nevertheless, the analytical solutions are still

limited even for these simplified models, so numerical

solution is the target of this research. This simplification

creates an error which needs to be overcome.

Diffusion waves neglect the acceleration terms, and kinematic

waves neglect both the pressure and the acceleration terms, in

the momentum equation. The kinematic wave model

represents unsteady flow through the continuity equation

while it substitutes a steady uniform flow for the momentum

equation [5]. A kinematic wave does not subside or disperse

as it travels downstream while it changes its shape. Kinematic

waves may be preferred in simulations of the natural flood

waves in steep rivers with slopes greater than 0.002 [6].

Diffusion occurs most in natural unsteady open channel flows

and in overland flow [5, 7, 8]. Diffusion waves may be

preferred in simulations of the flood waves in rivers and on

flood plains with milder slopes. There have been many studies

in the literature to solve the kinematic and diffusion wave

equations with several numerical methods [911] but it is good to find which numerical scheme more accurate,

convergence and stable. For this reason present work will

address computational aspects of solving the SaintVenant equations.

3- GOVERNING EQUATIONS, INITIAL AND BOUNDARY CONDITIONS

Diffusion wave equation is written as below:

In this equation Q is volume flow rate (Discharge ) , is the diffusion coefficient and C is the celerity of diffusion wave

and is calculated from the equation below:

C =

(2)

In the equation above w is the width of water surface and h is

the depth of water. In the diffusion wave equation, the time

derivative is from the first order, therefore for solving it, an

initial condition is needed. Also the spatial derivative is from

the first and second order and a boundary condition is needed

too. The first amounts related to flow are assumed for the

initial condition and written as below:

Q(x, 0) = 0 x L (3)

The Q(x, 0) is function of space x . There are different

figures to consider the boundary conditions. But normally

input hydrograph is assumed for the upstream boundary

condition:

Q (0, t) = (4)

and are functions of time and space respectively.

4- DESCRIPTION OF THE DIFFERENTIAL

QUADRATURE METHOD

In the Differential Quadrature method for estimating the n

degree derivative of the function in the direction of x in each

interval (a,b), the interval is divided to N parts see Fig.1

Fig. 1. Interval divided to N points for calculation of derivative of point i

Then derivative of function is calculated from the equation

below:

( ) i = 1,2,, (5)

In that equation, ( ) shows the amount of function in the

point of xj and is indicant of weight coefficient of each

point that shows the effect of point j in n time derivative

calculation of point (i) (Shu, 2000) .

Choosing the test function and appropriate grid points

distribution are tow effective parameters in determination of

the weight coefficients. For choosing the test function

polynomials test functions or harmonic test functions can be

used. Also there are two types of distribution of points that can

be used, uniform distribution and the Chebyshev-Gauss-

Lobatto cosine distribution that respectively is calculated in

the following equations:

(6)

* (

)+ (7)

Where, N number of grid points in domain direction ,L is the

length of domain .



4- DISCRETIZATION OF DQM IN CONJUNCTION WITH FDM

By applying the rule of Quadrature Method and finite

deference on Eq .(1), the linearized St. Venant equation is

converted to:

{

}

,

-

,

-

(8)

When the DQ method is used in time direction, for more

efficiency, the calculation field in t direction is divided to

several time blocks and the Eq. (1) in the time block of rb

written like below:

i,j = 1, 2,.,N

s = 2, 3,..., rb = 1, 2, 3,.,

N is the number of points in network in direction of space , is the number of points in the direction of time and Nb is the

total number of time block. In usage the DQ method, the

convective acceleration term is discretise as below:

,

-

,

-

(9)

In this equation the shows the weight coefficient of

differential quadrature, in the direction of x for each block .

Also for estimation of the time derivative the following

equation can be used:

,

-

{

}

(10)

, in t direction for each time step. In the

equation above, the initial condition for the first time block in

the same as the initial condition for the problem at the

beginning and for the other blocks the initial condition can be

achieved from the previous time block.

By substituting Eqs. (9) and (10) in Eq.(11) , the equation of

diffusion wave would be:

{

}

{

}

,

-

,

-

(11)

Rearrange Eq.(11) to get the unknowns .

{{

}

,

-

,

- }

(12)

And then Eq. (12) has been applied in the formulation of the

matrix at i (space) =3 and time s=2,3,4 R .

{

{

}

[

[

]

[

]

]

[

]

[ ]

[

]

* [ ] *

+-

[ ]

}

The boundary condition in Eq. (13) is written in the matrix

form as:

[ ]

[

]

(14)



The initial condition in Eq.(13) is written in the matrix form

as:

[ [ ] [ ]-

[ ]

The previous matric Eq.(13) is solved for the and which are unknown values. If the equation is used for a time

block that consists of points, in attention to the initial condition that is known for each time block, there

would be unknown. After writing the Eq.(13) for all of the internal point of each block, there would be

number of non-linear equation.

Fig. 2. Domain of the solution (one block)

The differential quadrature weighted coefficients can be

solved by using several techniques like Bellman's first

approach, Shu's general and Quan Chang's approach , in the

present study, Shu's general approach which is a numerical

discretization technique that delivers logical and accurate

numerical solution is used . Shu's general approach is based on

(Legendre polynomials) then the weight coefficient matrix can

be written in general as (Shu et al. 2004):

Where,

By substituting Eq.17 in Eq.16 yields ;

In this work, the weight coefficient matrix can be written as

(Shu et al. 2004):

( )

( )

*

+

A matrix of time domain can be written like B matrix in space

domain as:

Note that the Legendre interpolation shape functions L j (x)

have the following properties.

L j (x) = 1 if i = j (24)



5- EVALUATING OF THE MODEL For evaluation of the differential quadrature method in solving

the diffusion wave equation, a numerical example is used

which is prepared using MATLAB software.

In the numerical tests, once the combination method of DQ-

FD method was used; in order to discretization the location

derivative in the differential quadrature method was used and

for the time derivative the Finite difference on coming method

was used and next time the derivatives in the location and time

was estimated from the differential quadrature method. Then

the results from this study is compared with the numerical

schemes available in literatures .

6- APPLICATION TO CASE PROBLEMS In order to examine the coupled FD and DQ method , the

numerical example is adapted from published paper (K.

Bajracharya, D.A. Barry, 1997) in which upstream boundary

condition is estimated as:

(

)

The initial condition was assumed as: = 0. As well as the values of the celerity, c, and the diffusion coefficient ,D,

were taken as 1 m/s and 100 m3/ s, respectively. So the length

of channel under study is 5000 m and the simulation time is

10000 second.

7- CONVERGENT STUDIES IN THE DQ METHOD The aim of the convergent studies is to find out the minimum

using grid points in the DQ method, so that if we increase the

grid points more, the result would not be changed too much.

Fig.3 and Fig.4 show the convergence of coupled FD and DQ

method in the direction of space and time; for this purpose the

maximum calculated flow for every number of points is

drawn; the max. and min. amount of convergence is equal to

3.1114 and -0.2837 cubic meters per seconds respectively . As

it was shown, with the coupled FD and DQ method in the

space direction points become low (N = 7). Also for study the

convergence in the time direction, maximum flow was

calculated for each block and the number of points in each

block were drawn in the Fig.3; as it was mentioned in the

previous researches, when a few block is used, more points are

needed to reach the convergent, in other hand when more

blocks are used, less points need to get convergent. For

example, in 8 time block, minimum points are 13, and if we

use 16 blocks, the results in 7 points in each block get

convergent.

Fig. 3. Study of convergent for Max. flow in the space direction

Fig. 4. Study of convergent for Max. flow in the time direction

344344.5

345345.5

346346.5

347347.5

348348.5

349

0 5 10 15 20 25

Max

. flo

w (

m3 /

se

c(

Number of grid points in the space direction

4 TIME BLOCKS

8 TIME BLOCKS

16 TIME BLOCKS

24 TIME BLOCKS

BENCHMARK CASE FLOW RATE

346

346.5

347

347.5

348

348.5

0 5 10 15 20 25 30

Max

. flo

w (

m2 /

se

c(

Number of grid points in the time direction



8- CASE RESULTS AND DISCUSSION For evaluation of the coupled FD and DQ method in solving

this problem, the Crank Nicolson and McCormack schemes

presented for testing the new method. The result of this study

was shown in Fig.5, as it can be seen the output hydrographs

calculated in the both method are adopted so much with

hydrographs resulted from the Crank Nicolson and

McCormack solution. But in the calculation, the combination

method is sensitive to the time distribution and the long

simulation time makes the problem unstable. Therefore in

choosing the time and the distance distribution, stability

measures should be controlled. The equation which is usually

used for determination of stability is number of Current that

for supplying stability this number should be less than one.

But in using the coupled FD and DQ method in the both time

and space direction, in the choosing the simulation time, there

is no limitation and also with few points in the network, the

good results with accuracy is obtained. But in the combination

method to achieve an accurate result, more points in the

network must be calculated and so the time needed for

calculation would increase.

For the comparison of the presented method and other

numerical methods the norm. error defined as below:

[

]

(25)

Fig. 5. Coupled FD and DQ solution ,discharge hydrograph for the numerical example at x = 490m

Table 1 shows the comparison of the FD-DQ method with

other numerical methods. In comparison with other methods,

it should be mentioned that in the research done by the (K.

Bajracharya, D.A. Barry, 1997) and colleagues (2005) the

McCormack method introduced as the method with better

efficiency. With comparing the norm. error in the DQ and

McCormack methods consider that there is not too much

difference in the amount. But the McCormack method is an

explicit method and stability conditions in that should be

controlled to keep the Courant number less than one. For this

issue, the simulation time must be chosen shorter. Whereas the

coupled FD and DQ method, without using Courant number, it

would be still stable.

Table I comparison of norm. error in different methods

Numerical method FD-DQ

method

*McCormack

scheme

**DQM

Norm. Error 0. 96009498 1.3084 1.259489

*methods that were used by (K. Bajracharya, D.A. Barry, 1997)

** From previous study done by the researchers

-50

0

50

100

150

200

250

300

350

400

0 2000 4000 6000 8000 10000 12000

Q m

3/s

Time in seconds

Calculated DQM

crank Nicklson , benchmark case

MaCckomack

Coupled DQ with FD method



9- THE EFFECT OF THE DQ STRUCTURE As it was mentioned before, choosing an appropriate test

function for calculation of derivative coefficient and

distribution type of the points in the domain are two important

factors in applying the DQ method; for calculation of the

coefficients, harmonic function could be used (based on the

Fourier series expansion) or the polynomial functions (based

on the Lagrange interpolation) and for the grid points

distribution in the network can also be used uniform or cosine

distribution.

In this research, behavior of both functions in both forms was

studied. In these studies observed that the results in the cosine

distribution is more accurate than the uniform distribution.

Also the results from the harmonic function and the

polynomial function dont have any sensitive difference. Fig. 6 shows the estimated output hydrograph with using different

ways.

Fig. 6. Effect of grid distribution on the solution., x = 490 m

10- CONCLUSION In this research, a novel approach is proposed for solution of

the diffusion wave model. Coupled Deferential Quadrature

and finite deference method was used to solve the one-

dimensional Diffusion wave model. The study in this research

shows that the coupled Deferential Quadrature and finite

deference method can be used for the solving the diffusion

wave model with using the least points in the network and has

good results; also this method has good efficiency and

stability in results presentation. In applying the coupled

Deferential Quadrature and finite deference method, the

results in the time and space direction are more accurate than

using the DQM . Also, as it was shown in the previous

researches, with dividing the calculation field in the time

direction to several time blocks, the calculation period

decrease.

The DQ method for solving the diffusion wave equation is not

sensitive to choosing the test function and the results has the

same accuracy. But distribution type of the points in the

network has magnificent effect in the results; so, the results

from cosine distribution are more accurate than the results

from the uniform distribution.

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0

50

100

150

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350

400

0 1000 2000 3000 4000 5000 6000 7000

Q (

m3/s

)

Time in seconds

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