Applied Mathematical Modelling 31 (2007) 1594–1608

www.elsevier.com/locate/apm

A differential quadrature analysis of unsteadyopen channel flow

Mohammad R. Hashemi a, Mohammad J. Abedini a,*, Parviz Malekzadeh b

a Department of Civil Engineering, Shiraz University, Shiraz, Iranb Department of Mechanical Engineering, Persian Gulf University, Bushehr, Iran

Received 1 June 2005; received in revised form 1 April 2006; accepted 11 May 2006Available online 17 July 2006

Abstract

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation ofunsteady open channel flow. To the best of authors’ knowledge, this is the first attempt to use the DQM in open channelhydraulics. The Saint-Venant equations and the related nonhomogenous, time dependent boundary conditions are discret-ized in spatial and temporal domain by DQ rules. The unknowns in the entire domain are computed by satisfying governingequations, boundary and initial conditions simultaneously. By employing DQM, accurate results can be obtained using dra-matically less grid points in spatial and time domain. The stability of DQM solution is not sensitive to choosing time step orCourant number unlike other methods. Although numerical problems such as instability, oscillation and underestimationnear critical depth can be seen by using other methods but DQM solution is smooth and accurate in this case. The results aresensitive to grid distribution in time domain. In light of this, Chebyshev–Gauss–Lobatto distribution performance is excel-lent. To validate the DQM solutions, the obtained results are compared with those of the characteristic method. In conclu-sion, DQM is a potential powerful method with minimum computational effort for unsteady flow simulation.� 2006 Elsevier Inc. All rights reserved.

Keywords: Saint-Venant equations; Stability; Courant number; DQM; Unsteady flow

1. Introduction

The simulation of unsteady open channel flow by solving the Saint-Venant equations is an essential step formany applied problems such as flood forecasting, dam break analysis and watershed modeling.

As early as development of computers, various numerical methods have been used for numerical modeling ofunsteady flow. Method of characteristics (MOC) due to its ability to describe physical flow processes and bound-ary conditions has received special attentions [1]. Despite many advantages of MOC, it requires a lot of gridpoints and much computational effort to yield a convergent accurate solution. The MOC is also sensitive toproper choice of grid spacing for stability and is more complicated to program, compared to the other methods.

0307-904X/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2006.05.006

* Corresponding author. Fax: +98 711 6286619.E-mail address: abedini@shirazu.ac.ir (M.J. Abedini).

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1595

Finite difference is another popular method which has been widely used. Although implicit finite differencemethods are claimed to be unconditionally stable but due to nonlinearity of the Saint-Venant equations, theyhave stability problems in practice and are highly sensitive to the time step. Many professional softwares devel-oped based on implicit finite difference methods (such as MIKE-11 and HEC-RAS) use Courant-number con-dition for time step to avoid instability problems [2,3]. In recent years, finite volume methods have significantprogress in simulation of complex unsteady flow phenomena such as dam break and transcritical flow [4,5]. Thismethod, has also been extensively employed in the research works but is not as efficient as other methods in prac-tical hydraulic engineering problems [6]. In addition to the above popular methods, some other new methodssuch as semi-analytical method have been introduced to deal with the Saint-Venant equations [7].

Besides all these works, researchers are always seeking new numerical methods with more efficiency andaccuracy to promote numerical modeling capabilities. In 1971, Bellman [8,9] introduced a new computationalmethod, so called, differential quadrature (DQ). Following Bellman’s research, DQ was used in diverse areasof computational mechanics and has been claimed as a highly accurate with minimum computational costmethod by many researchers. Bert and Malik [10] in their review article covered a complete discussion of chro-nological development of the method. This method was successfully applied in structural analysis [11–17] andheat transfer [18]. Shu published several papers about generalized DQ for numerical simulation of Navier–Stokes (N–S) equations and boundary layer in the field of fluid mechanics [19–22].

Application of DQM for solving transient or time dependent problems is another challenging problem incomputational fluid or solid mechanics. Among several articles which have been published to solve N–S equa-tions using DQM, a few are limited to steady state condition [23] while some others have implemented DQ todiscretize spatial domain and either finite difference or Runge Kutta method to discretize the time domain [24].Using DQM in spatial domain and other methods in temporal domain (hybrid method) is also common prac-tice in solid mechanics [25]. An important drawback of hybrid methods is instability problems which havebeen reported in the previous studies [26]. Recently, implementation of DQM for discretization of the tempo-ral parts of the transient problems have been considered in a few works [27,28]. Block marching in time is anefficient method to solve time dependent problems in this respect [29].

Based on the above discussion, although DQ have been implemented by several researchers for solving N–Sequations in computational fluid dynamics, but it seems that the capabilities of the method for simulation offree surface flow in open channels have not been explored yet.

In this work, DQM is examined as an alternative numerical method for unsteady flow modeling. At first,the Saint-Venant equations are discretized in both spatial and temporal direction by DQM. The state variables(i.e., discharge and depth) are computed by solving the resulting system of nonlinear equations in the entirecomputational domain. As early results of this study shows, the simulation of the problem by DQM in onestep requires relatively large number of grid points in time zone to capture the boundary condition precisely.Consequently, the computational time will start to grow. To handle this problem, DQM was used in successiveincrements in the time zone. The performance of DQM is demonstrated via a numerical example. The resultsare compared with other traditional numerical methods in order to explore merits and drawbacks of DQM.Finally some conclusions based on this study are cited.

2. The governing equations and the related boundary conditions

Saint-Venant equations are the most common equations which are being used for numerical simulation ofone-dimensional free surface flow in open channels and rivers. These equations can be directly derived by inte-grating Navier–Stokes equations over the cross section. Among different forms of the state variables [30], herewe adopt discharge and depth as state variables. Accordingly, the Saint-Venant equations can be written as

oU

otþMðUÞ oU

oxþ SðU; x; tÞ ¼ 0; ð1Þ

in which U, M and S are given by

U ¼h

Q

� �; M ¼

D E

F G

� �; S ¼

H

I

� �; ð2Þ

1596 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

where h denotes depth of water with respect to channel bed and Q flow discharge. The elements of thecoefficient matrix M, and source term vector S, are functions of the state variables and channel properties,i.e.,

TableDiffere

Upstre

Q(0, t)h(0, t)FU(Q(

D¼ 0; Eðh;QÞ¼ 1

BðhÞ ; F ðh;QÞ¼ gAðhÞ�BQ

AðhÞ

� �2

; Gðh;QÞ¼ 2Qðh;QÞAðhÞ ;

Hðh;Q;x; tÞ¼�qðx; tÞBðhÞ ; IðQ;h;x; tÞ¼� Q

AðhÞ

� �2oAðhÞ

ox

�����h¼cte:

�gAðhÞ S0�n2 QjQjP ðhÞ4=3

AðhÞ10=3

!�qðx; tÞV L cosU;

ð3Þ

where B is the top width, g acceleration of gravity, S0 bed slope, q lateral flow discharge, VL lateral flow veloc-ity, U angle of lateral flow with the direction of main flow, n the Manning coefficient and P wetted perimeter. Itshould be noted that the first and second row of Eq. (1) are called the continuity and momentum equations,respectively. In addition to governing equations, the initial and boundary conditions must be specified for thenumerical simulation of flow.

The initial values of the state variables are given at the start of the simulation as initial conditions, whichcan be stated as

Uðx; 0Þ ¼h0ðxÞQ0ðxÞ

� �; ð4Þ

where h0 and Q0 are known functions of x. Most often, the steady state water surface profile for initial dis-charge may be used for initial condition.

The boundary conditions for the Saint-Venant equations depends on the regime of the flow. For supercrit-ical flow, two upstream boundary conditions must be imposed at upstream boundary and none in the down-stream while in subcritical regime one condition is required at upstream and another in the downstream of thechannel. In this paper, subcritical flow is considered in detail but DQM can be formulated easily for super-critical flow (see Appendix B for more detail). Table 1 represents three types of boundary conditions atupstream or downstream of a channel for unsteady subcritical flow . These include known discharge or depthas a function of time and discharge–depth relationship. In general, for the practical unsteady flow problemsdiscretized observed data are available as boundary conditions which cannot be expressed in closed forms.However, for special regular channel cross sections some closed form functions may be used to describe dis-charge–depth relationship. Although any type of these boundary conditions may be applied for each bound-ary but known discharge at upstream which imply flow hygrograph and discharge depth relationship at thedownstream of the channel are the most common boundary conditions in practical problems, which can beexpressed as

Qð0; tÞ ¼ QUðtÞ; F DðQðL; tÞ; hðL; tÞÞ ¼ 0: ð5Þ

For example, if a channel terminates in a free overfall or a drop, it may be assumed that, the critical depth willform at this boundary which may be written as

F DðQðL; tÞ; hðL; tÞÞ ¼ QðL; tÞ � AðL; tÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig

AðL; tÞBðL; tÞ

s¼ 0: ð6Þ

1nt possible forms for the Saint-Venant boundary conditions

am Downstream

= QU(t) Q(L, t) = QD(t)= hU(t) h(L, t) = hD(t)0, t),h(0, t)) = 0 FD(Q(L, t),h(L, t)) = 0

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1597

It should be noted that in critical depth condition, the flow velocity is equal to the wave celerity. Consequently,numerical instability may occur near or at the critical depth. A few investigators use special treatment to han-dle this problem [2].

3. DQ solution for unsteady flow

As was stated before, in many previous applications of DQM for unsteady or dynamic problems, the DQMwas only used in the spatial domain and finite difference or Runge Kutta methods were incorporated for dis-cretization of the temporal derivatives. In this research, DQ rule is directly used for both space and time deriv-atives. The implementation of DQ in the time direction needs a special treatment to guarantee the efficiency ofthe method. The detail implementation of DQM to the Saint-Venant equations is presented here.

Contrary to other conventional transient schemes where computation of state variables is achieved viamarching in time, in classical DQM computation of state variables over the entire x–t plane occurs at one step.In many unsteady flow problems such as flood forecasting, tidal river simulation and so on, the simulationtime may be a long interval of time. Besides, the time steps between water depth or flow records which areused as boundary conditions may be very small as compared to the total simulation time. Therefore, one mustuse very short time step (which is equivalent to large number of grid points in the time direction in using globalDQM) to detect boundary conditions precisely. In the meanwhile, as the number of grid points increases (notfor convergence but for capturing the boundary conditions), the order of the system of equations which mustbe solved to compute the state variables dramatically increases and so the computational time will start togrow. The errors due to weighting coefficients also increases.

In order to overcome these drawbacks, DQM can be used in several time blocks or increments. The similarmethod has been previously applied for solving transient problems in the continuum mechanics [27,29]. Inother words, one can split total simulation time into a number of smaller ones and use the final solutionsof each time increment for the next time increment (see Fig. 1), which may be expressed as

Urþ1ðxi; t1Þ ¼ Urðxi; tNrtÞ or

hrþ1i1

Qrþ1i1

!¼

hriNr

t

QriNr

t

!; ð7Þ

Fig. 1. Computational domain for DQM in several time increments.

1598 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

where r denotes the increment number and Nrt the total number of time grid points in the rth increment. This

method effectively handles above-mentioned drawbacks.For a typical increment r and for an arbitrary point inside the computational domain, with xi and tr

k as itscoordinates, the discretized form of Eq. (1) become

oU

ot

� �r

ik

þMðUrikÞ

oU

ox

� �r

ik

þ SðUrikÞ ¼ 0 for i ¼ 2; 3; . . . ;Nx � 1; k ¼ 2; 3; . . . ;Nr

t ; r ¼ 1; 2; . . . ;NIC:

ð8Þ

The time and space derivatives are expanded using conventional DQ rule. Accordingly, for the time derivativeone has

oU

ot

� �r

ik

¼

ohot

� �r

ik

oQot

� �r

ik

0BBB@

1CCCA ¼

PNrt

n¼1

CT rknhr

in

PNrt

n¼1

CT rknQr

in

0BBB@

1CCCA; ð9Þ

where CT r represents the DQ weighting coefficients for the time direction corresponding to increment r. Theseweighting coefficients are known for a specified grid distribution. For more detail concerning computation ofweighting coefficients, see Section 4.3. The time derivative may be expanded in more detail as

oU

ot

� �r

ik

¼

PNrt

n¼2

CT rknhr

in

PNrt

n¼2

CT rknQr

in

0BBB@

1CCCAþ CT r

k1hri1

CT rk1Qr

i1

� �: ð10Þ

Since, the initial values of the state variables are given at the beginning of each increment, they can be added tothe source terms as follows:

S�rik ¼ Srik þ

CT rk1hr

i1

CT rk1Qr

i1

� �: ð11Þ

Similarly, the space derivative can be discretized by using DQ rule as

oU

ox

� �r

ik

¼

PNx

m¼1

CX rimhr

mk

PNx

m¼1

CX rimQr

mk

0BBB@

1CCCA; ð12Þ

where CX denotes, the DQ weighting coefficient in the x direction. Finally, the discretized form of the govern-ing equations become,

grðxi; tkÞ ¼gr

11ðxi; tkÞgr

12ðxi; tkÞ

� �¼

PNrt

n¼2

CT rknhr

in

PNrt

n¼2

CT rknQr

in

0BBB@

1CCCAþ 0 Er

ik

F rik Gr

ik

� ��

PNx

m¼1

CX rimhr

mk

PNx

m¼1

CX rimQr

mk

0BBB@

1CCCAþ H r

ik þ CT rk1hr

i1

I rik þ CT r

k1Qri1

� �¼ 0

for i ¼ 2; 3; . . . ;Nx � 1; k ¼ 2; 3; . . . ;Nrt ; r ¼ 1; 2; . . . ;NIC: ð13Þ

Eq. (13) gives 2ðNrt � 1ÞðN x � 2Þ nonlinear simultaneous equations. The total unknowns, for increment r, can

be stated as

Urik for i ¼ 1; 2; . . . ;Nx; k ¼ 2; 3; . . . ;Nr

t ; ð14Þ

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1599

which are 2ðNrt � 1ÞðNxÞ unknown variables. Obviously, 4ðN r

t � 1Þ extra equations are needed to compute theunknowns. To obtain sufficient number of equations, the boundary conditions should be implemented. Thiswill be discussed in the next section.

3.1. Implementation of boundary conditions

Refereing to Eq. (5), assuming known discharge at upstream boundary (i.e., i = 1), the discretized form ofboundary condition become

Qr1k � Qr

UðtkÞ ¼ 0; k ¼ 2; 3; . . . ;N rt ; ð15Þ

where QrU as mentioned before, is a known function of time for each increment. Similarly, downstream bound-

ary condition (i.e., i = Nx) can be written as

F DðQðL; tÞ; hðL; tÞÞ ¼ F rDðQr

Nxk; hrNxkÞ ¼ 0; k ¼ 2; 3; . . . ;Nr

t : ð16Þ

F rD is a known depth–discharge relationship. As an example, the critical water depth (Eq. (6)) in a rectangular

channel (i.e., B = b and A = bh) may be written as

QrNxk � b

ffiffiffigp ðhr

NxkÞ1:5 ¼ 0; k ¼ 2; 3; . . . ;Nr

t : ð17Þ

By implementing boundary conditions, 2ðNrt � 1Þ equations are added to global system of equations for each

increment. Therefore, as discussed in the previous section, still 4ðNrt � 1Þ � 2ðNr

t � 1Þ ¼ 2ðN rt � 1Þ equations

are required. Eq. (1) can be considered as two hyperbolic first-order partial differential equations. Therefore,each equation needs only one boundary condition. Based on the above boundary conditions, since, Q is givenat the upstream boundary, then, the continuity equation should be satisfied at the other grid points includingdownstream boundary, i.e.,

gr11ðxNx ; tkÞ ¼ 0; k ¼ 2; 3; . . . ;N r

t : ð18Þ

The momentum equation should be satisfied at the upstream boundary in a similar manner, i.e.,

gr12ðx1; tkÞ ¼ 0; k ¼ 2; 3; . . . ;N r

t : ð19Þ

Obviously, by adding Eqs. (18) and (19) to the system of equations, enough number of equations are formed.This nonlinear system of equations can be solved for 2ðNr

t � 1ÞðN xÞ unknowns which are discharge and depthvalues in each increment. Further detail regarding method of solving these equations can be found in Appen-dix A. The performance of DQM may be better shown via numerical experiment as presented in the nextsections.

4. Numerical example

In order to assess the computational efficiency and convergence of DQM and also to validate the solutions,a numerical example is considered.

Example. A rectangular channel 1000 m long and 6 m wide is laid on a slope of 0.0064. The Manning n for thechannel is 0.03416 and the channel terminates in a drop (i.e., critical depth condition for downstreamboundary). A time varying discharge enters at the upstream of the channel is shown in Fig. 2. The initial depthand discharge in the channel are 1.02296 m and 12 m3/s, respectively. It is desired to compute the time historyof water depth at x = 500 and x = 800 for total simulation time, 12000 s.

By employing a suitable numerical method, the time history of water depth and discharge can be computedfor each point in the channel. This example is solved by DQM, the diffusive scheme and the method of char-acteristics (MOC). At first, a convergence study for DQM results is presented and next the results are com-pared with other methods.

0 2000 4000 6000 8000 10000 1200010

15

20

25

30

35

40

Time, Sec

Dis

char

ge,

m3 /s

Fig. 2. Upstream discharge hygrograph for the numerical example.

1600 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

4.1. Convergence study of the solution

Numerical experiments using different values for Nx and N rt for the above example, show that unlike the

traditional methods, the stability of the DQM solution is not dependent on the time step or Nrt . However,

the magnitude of the results changes slightly as the number of grid points in the time domain changes. In orderto find out the sufficient number of grid points for the problem, a convergence study is carried out.

For a typical point of the channel located at x = 800 which is near downstream boundary, the time historyof discharge and depth are computed after each run (see Fig. 7). Fig. 3 shows the maximum computed depth

5 10 15 20 25 30 35 40 45 50 551.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

Ntr

Max

. Co

mp

ute

d D

epth

, m

Number of Increments=3Number of Increments=6Number of Increments=1Converged Value

Fig. 3. Convergence study of maximum computed depth at x = 800.

Table 2Effects of Nr

t and number of increments on the computational time

Number of increments(NIC)

Nrt hmax CPU

time (s)

1 15 2.20 3139 2.25 31159 2.26 865

3 13 2.25 4017 2.26 6821 2.27 110

6 5 2.25 127 2.26 309 2.27 38

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1601

(peak point) for different values of N rt at this point. Obviously, the sufficient number of Nr

t for DQM dependson the number of increments. In other words, as the number of increments increases the required number ofgrid points in each increment (i.e., Nr

t ) decreases. As can be seen in this figure, the converged value of maxi-mum depth at x = 800 m is 2.27 m. For 3 increments at least 21 grid points are required while for 6 increments9 grid points is sufficient to yield converged solution. It is interesting to note that using very few gird points inthe time domain also results in values which are very close to desired solution (for Nr

t ¼ 3 the maximum erroris 5 cm (2.2%)).

Table 2 shows the effects of number of time increments and Nrt on the computation time. As this table

shows, using small values for N rt and increasing the number of increments is more efficient than using large

values for Nrt and reducing the number of increments. For example, using N r

t ¼ 9 and 6 time increments orusing N r

t ¼ 21 and 3 time increments converge to the same result for maximum computed depth at x = 800(i.e., hmax = 2.27). However, the computational time for 6 time increments is significantly smaller. Regardingthe sufficient number of grid points in the x direction, it should be noted that, although a few number of gridpoints may yield solutions with acceptable error, at least 11 grid points must be used to yield a convergentsolution. Fig. 4 shows a typical convergence study for Nx. As can be seen in this figure, for 11 grid pointsin the x direction the maximum computed depth at x = 500 will converge to 2.29 m.

4 6 8 10 12 14 16 182.23

2.24

2.25

2.26

2.27

2.28

2.29

2.3

Nx

Max

. Co

mp

ute

d D

epth

, m

Fig. 4. Convergence study of maximum computed depth at x = 500.

300 320 340 360 380 400 420 440 460 480 500

1500

2000

2500

3000

3500

4000

4500

5000

5500

Tim

e, s

Distance, m

Fig. 5. Grid of characteristics for a small typical portion of the computational domain.

1602 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

4.2. Comparison of DQM results with MOC and FDM

Since, there is no analytical solution for general nonlinear unsteady flow equations, the grid of character-istic (GC) method is used as benchmark to validate the solution. Accordingly, a GC-based program is writtenfor this purpose. Based on the previous researches on the MOC, this method has special physical and numer-ical attractive features and is generally claimed to be more accurate than other methods [1]. For brief, thedetail of formulation of MOC is not presented here which can be found elsewhere [1,31]. The diffusive scheme,which is a traditional explicit finite difference scheme [32] is also used for comparison in other aspects such ascomputational time and number of grid points. It should be noted that in explicit finite difference methods, thestate variables at boundary points must be computed with the aid of MOC.

Fig. 5 represents irregular grid of characteristics for this numerical example. This figure shows that a lot ofgrid points must be generated to obtain an accurate solution by MOC. The computed water depth based onMOC, DQM and diffusive scheme are plotted at x = 500 and x = 800 in Figs. 6 and 7. These figures show thatDQM solution is in excellent agreement with the water depth computed by the MOC. The diffusive schemeunderestimate the water depth specially near the critical depth or downstream boundary (i.e., at x = 800).

Despite the accuracy of MOC solution, some degree of oscillation in its results specially at the peak pointcan be seen. As one approaches the critical depth boundary, the oscillatory nature of MOC solution increases.This fact can be observed by comparing computed depth at x = 800 which is near the downstream boundarywith results obtained at x = 500. The nonphysical oscillation near steep gradients which is produced by con-ventional schemes was also reported by other investigators [5]. On the other hand, the simulated results forDQM are shown to be smooth and monotone, which is an important advantage of DQM.

Table 3 represents a comparison between different methods in terms of computational efforts. As this tableshows, DQM with several time increments requires the least number of grid points (i.e., 2 · 9 · 6 · 11 = 1188)and computation time compared to the other methods.

4.3. Effects of grid point distribution and basis functions

There are two key factors for successful implementation of DQM. One, the choice of grid point distribu-tions or sampling points and two, the type of test functions, which are used for computation of the weightingcoefficients. These factors are main parts of DQM structure and were well discussed in the previous works (for

0 2000 4000 6000 8000 10000 120000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Time, Sec

MOCDiffusive schemeDQM

4000 4100 4200 4300 4400 4500 4600 4700

2.24

2.25

2.26

2.27

2.28

2.29

2.3

2.31

Dep

th, m

D

epth

, m

Time, Sec

MOCDiffusive schemeDQM

a

b

Fig. 6. Depth versus time at x = 500. (a) Full solution, (b) magnified solution near the peak.

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1603

e.g., see [10,22]). In this research, sensitivity of DQM solution to uniform and Chebyshev–Gauss–Lobatto(cosine) grid distributions for both polynomial and harmonic (sinusoidal) test functions are investigated.The weighting coefficients for polynomial and harmonic basis functions are computed using Shu’s generalapproach [33,34].

Fig. 8 shows a sample result for this study using different distributions and basis functions. As this figure dem-onstrates, computed depths based on the uniform distribution deviate from accurate solution in some parts, spe-cially at the start of simulation. Furthermore, the performance of harmonic basis functions are better thanpolynomial functions for this distribution. Contrarily, the cosine grid distribution can produce accurate resultsfor both types of basis functions. In summary, the results of several numerical experiments with different numberof grid points in spatial and time domain using different basis functions and grid distribution indicate that:

• The solution is not sensitive to gird point distribution in x direction.• For uniform grid distribution in time domain, the harmonic test functions produce more accurate results

compared to polynomial test functions.

0 2000 4000 6000 8000 10000 120000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Dep

th, m

Time, Sec

MOCDiffusive schemeDQM

4000 4200 4400 4600 4800 5000

2.1

2.12

2.14

2.16

2.18

2.2

2.22

2.24

2.26

2.28

Dep

th, m

Time, Sec

MOCDiffusive schemeDQM

a

b

Fig. 7. Depth versus time at x = 800 near critical depth boundary. (a) Full solution, (b) magnified solution near the peak.

Table 3Comparison of computational efforts for different numerical methods

Method Nx Nrt NIC Total degrees of freedom Comput. time (s)

MOC 21 – 1 48012 653Diffusive scheme 21 1800 1 75600 205DQM 11 59 1 1298 634

11 9 6 1188 38

1604 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

• For cosine grid distribution in the time domain, both harmonic and polynomial test functions yields accu-rate results.

• In general, cosine grid distribution shows much better performance compared to uniform grid distribution.

0 2000 4000 6000 8000 10000 120000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Dep

th, m

Time, Sec

Benchmark SolutionDQM (Harmonic, Uniform)DQM (Polynomial, Uniform)DQM (Polynomial, Cosine)

200 300 400 500 600 700 800

0.95

1

1.05

1.1

1.15

Dep

th, m

Time, Sec

Benchmark SolutionDQM (Harmonic, Uniform)DQM (Polynomial, Uniform)DQM (Polynomial, Cosine)

a

b

Fig. 8. Effect of test functions and grid distribution on the solution. (a) Full solution, (b) magnified solution near the beginning.

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1605

5. Conclusion

DQM can be effectively used for numerical simulation of unsteady open channel flow. An important advan-tage of DQM is its convergence without any limitation on the time step or Courant number as apposed toother traditional numerical methods. By using much less grid points specially in the time domain, accuratesolution can be obtained.

The computational time of DQM is the least compared to MOC and diffusive scheme.In order to obtain desired solution, it is more efficient to use a few number of grid points in each time incre-

ment (e.g., Nrt ¼ 9) and increase the number of increments rather than selecting large values for Nr

t and reduc-ing the number of time increments.

Although numerical oscillation can be seen near critical depth boundary, using the traditional methodssuch as MOC, the performance of DQM is smooth and monotone in this case. This problem arises nearthe sharp water surface gradients such as drops.

1606 M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608

The number and the distribution of grid points in the time domain is crucial to achieve desirable solution.Between uniform and cosine distribution, cosine has better performance. As the number of grid pointsincreases, uniform grid distributions fails to converge. Although, DQM solution is less sensitive to type of testfunctions but for uniform grid distribution, harmonic test functions produce more accurate results comparedto polynomial test functions.

Evaluation of this method for more complicated problems such as nonprismatic channels, transcritical flowregime, and two-dimensional flow is underway and will be published subsequently.

Appendix A. Solution of system of nonlinear equations

The details of solving 2ðNrt � 1ÞNx system of nonlinear algebraic equations for each increment are presented

here. For simplicity a typical rectangular channel of width b is considered. Assuming no lateral discharge, thegoverning equations should be rewritten using the following elements for the coefficient matrix and sourceterms vector (see Eq. (1)) which gives

D ¼ 0; E ¼ 1

b; F ðQ; hÞ ¼ gbh� b

Qbh

� �2

; GðQ; hÞ ¼ 2Qbh; H ¼ 0;

Iðh;QÞ ¼ �gbh S0 � n2 QjQjðbþ 2hÞ4=3

ðbhÞ10=3

!: ð20Þ

The traditional Newton’s method is employed to solve the discretized form of the equations which may bewritten as

Xriterþ1 ¼ Xr

iter � fJrg�1iterðgr

iterÞ; ð21Þ

where Xr is unknown vector for increment r and subscript iter denotes the iteration number. The key point forimplementation of the method is computation of Jacobian matrix which has 2� ðNr

t � 1ÞNx columns for eachpoint (i.e., i,k) in the computational domain. The Jacobian matrix can be obtained by taking derivative of Eq.(13) with respect to unknowns, which are discharge and depth at each grid point, i.e.,

Jr ¼ ogrðxi; tkÞoUr

pq

¼ o

oUrpq

gs11ðxi; tkÞ

gs12ðxi; tkÞ

� �: ð22Þ

For instance, the elements of the jacobian matrix for the momentum equations can be computed by

Jðsrik;pþNxþ2Nxðq�2ÞÞ¼ ogs

12

ohrpq

¼�2Qrik

bhrik

2dipdkq

XN x

m¼1

CX rimQr

mkþdipdkq bgþ2Qrik

2

bhrik

3

!XNx

m¼1

CX rimhr

mk

þ bghrik�

Qrik

2

bhrik

2

!XNx

m¼1

CX rimdmpdkqþdipdkq

� �gbS0þgn2

b7=3hrik

10=3Qr

ikðbþ2hrikÞ

4=3

!

� gn2Qrik

2

b7=3ðbþ2hr

ikÞ1=3 10

3hr

ik�10=3þ4hr

ik�7=3

� �� �dipdkq;

i¼ 1;2; . . . ;Nx�1; k¼ 2; . . . ;Nrt �1; p¼ 1;2; . . . ;Nx; q¼ 2; . . . ;Nr

t

ð23Þ

and

Jðsrik; p þ 2N xðq� 2ÞÞ ¼ ogs

12

oQrpq

¼XNr

t

n¼2

CT rkndipdnq þ

2

bhrik

dipdqk

XNx

m¼1

CX rimQr

mk þ2Qr

ik

bhrik

XNx

m¼1

CX rimdmpdkq

� 2Qrik

bhrik

2dipdkq

XNx

m¼1

CX rimhr

mk þ2n2g

ðbhrikÞ

7=3ðbþ 2hr

ikÞ4=3Qr

ikdipdkq;

i ¼ 1; 2; . . . ;Nx � 1; k ¼ 2; . . . ;Nrt � 1; p ¼ 1; 2; . . . ;Nx; q ¼ 2; . . . ;Nr

t ; ð24Þ

M.R. Hashemi et al. / Applied Mathematical Modelling 31 (2007) 1594–1608 1607

where d is Kronecker delta function. The other elements of Jacobian matrix can be determined in the samemanner by taking derivative of discretized continuity and boundary condition equations with respect to dis-charge and depth at each grid point.

Appendix B. Formulation of DQM for supercritical flow

As noted in Section 2, for supercritical flow there are two upstream boundary conditions (i.e., at i = 1) andno downstream boundary condition (i.e., at i = Nx). In other words, discharge and depth are known forupstream boundary. Therefore one can omit left boundary unknowns (i.e., Q1k, h1k for k ¼ 2; 3; . . . ;Nr

t ) fromthe unknown vector. It should be noted that, the initial values of state variables are omitted from system of theequations in a similar manner. Accordingly, the number of unknowns become 2ðN r

t � 1ÞðN x � 1Þ for the entirecomputational domain (for subcritical flow there are 2� ðNr

t � 1ÞN x unknowns). On the other hand, the dis-cretized form of the continuity and momentum equations must be satisfied for i = 2,3, . . . ,Nx andk ¼ 2; 3; . . . ;N r

t which form 2ðNx � 1ÞðNrt � 1Þ equations too and can be solved for computing the total

unknowns.It must be noted that, the boundary values of the state variables will appear directly in the equations as one

expands space derivative by DQM. For example the space derivative of discharge can be expanded as,oQox ¼

PNxm¼1CX r

imQrmk ¼ CX r

i1Qr1k þ

PNxm¼2CX r

imQrmk. Hence, the boundary values of discharge (i.e., Q1k) has direct

contribution in the continuity and momentum equations.

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