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

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1 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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 , and 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 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.

2 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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 Saint Venant 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 [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 [9 11] 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 Saint Venant equations. 3- GOVERNING EQUATIONS, INITIAL AND BOUNDARY CONDITIONS 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). 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 = 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 (2) 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.

3 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: Rearrange Eq.(11) to get the unknowns. 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,., {{ }, -, - } (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. { { } 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) The boundary condition in Eq. (13) is written in the matrix form as: (14)

4 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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, A matrix of time domain can be written like B matrix in space domain as: By substituting Eq.17 in Eq.16 yields ; Note that the Legendre interpolation shape functions L j (x) have the following properties. In this work, the weight coefficient matrix can be written as (Shu et al. 2004): L j (x) = 1 if i = j (24)

5 Max. flow (m 2 / sec( Max. flow (m 3 / sec( International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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 m 3 / s, respectively. So the length of channel under study is 5000 m and the simulation time is 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 and 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 Number of grid points in the space direction 4 TIME BLOCKS 8 TIME BLOCKS 16 TIME BLOCKS 24 TIME BLOCKS BENCHMARK CASE FLOW RATE Fig. 3. Study of convergent for Max. flow in the space direction Number of grid points in the time direction Fig. 4. Study of convergent for Max. flow in the time direction

6 Q m 3 /s International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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) Calculated DQM crank Nicklson, benchmark case MaCckomack Coupled DQ with FD method Time in seconds 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 FD-DQ *McCormack **DQM Numerical method method scheme Norm. Error *methods that were used by (K. Bajracharya, D.A. Barry, 1997) ** From previous study done by the researchers

7 Q (m 3 /s) International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 14 No: 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 don t have any sensitive difference. Fig. 6 shows the estimated output hydrograph with using different ways FD-DQ METHOD(Polynomail-CGL distribution crank Nicklson, benchmark case FD-DQ METHOD (polynomial-uniform) Time in seconds 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 onedimensional 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. REFERENCES [1] Bellman R. Kashef B. G. and Casti J Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics. 10(1): [2] Chen C.N Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integrated numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes. Applied Mathematical Modeling. 24(8-9): [3] Chow V.T. Maidment D.R. and Mays L.W Applied Hydrology. New York: McGraw Hill. [4] Bajracharya, K. and Barry, D. (1997). Accuracy criteria for linearised diffusion wave flood routing. Journal of Hydrology, 195, pp [5] X. Ying, A.A. Khan, S.S.Y. Wang, Upwind conservative scheme for the Saint Venant equations, J. Hydraul. Eng. ASCE 130 (10) (2004) [6] Collier, N., Radwan, H., Dalcin, L., and Calo, V. M. (2011). Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach. Procedia Computer Science, 4, pp [7] T.C. Lackey, Sotiropoulos, Role of artificial dissipation scaling and multigrid acceleration in numerical solutions of the depthaveraged free-surface flow equations, J. Hydraul. Eng. ASCE 131 (9) (2005) [8] A. M. Wasantha Lal, (2012) Numerical Errors in Diffusion Wave Models When Simulating Kinematic Flow. World En vironmental And Water Resources Congress 2012.

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