DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION OF THE DIFFUSION WAVE MODEL

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1 JOURNAL OF FLOOD ENGINEERING JFE 1() July December 1; pp DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION OF THE DIFFUSION WAVE MODEL Birol Kaya, Yalcin Arisoy & Asli Ülke Civil Engineering Dept., Dokuz Eylul Uni., 3516, Buca, Izmir, Turkey ABSTRACT: The differential quadrature method (DQM) is used for the numerical solution of diffusion wave equation derived from the Saint-Venant equations for one-dimensional, gradually varied, unsteady open-channel flow, which is the newest attempt in open channel hydraulics. The results are compared with the performance of the finite difference method (FDM). Based on the comparison of both the explicit and implicit finite difference approximations to the solution, it is concluded that the DQM provides similar results but with higher speed, fewer nodes and less memory usage. Keywords: Differential quadrature method, Saint-venant equations, Diffusion wave, Unsteady flow. 1. INTRODUCTION The Saint Venant formula, written in 1848 for gradually varied unsteady flows in open channels, can not be solved analytically due to nonlinear constraints. In order to neglect some terms in the Saint Venant equations, they are linearized first by means of different wave approximations such as kinematic wave, non-inertia wave, gravity wave and quasi-steady dynamic wave, and then a numerical solution technique like the finite difference or finite element method is widely used. This simplification procedure has been reviewed systematically by Yen and Tsai (1) and Fan and Li (6). According to these studies, the first application of the diffusion analogy to flood routing originated from Hayami in After Hayami (1951), Appleby (1954), Cunge (1969), Dooge et al., (1987), Ponce (199), Rutschmann and Hager (1996), Sivalapan et al., (1997), Bajracharya and Barry (1997) and many others carried out similar studies in the following years. Singh (1996) gave a detailed definition for the diffusion wave equation application in hydrological modelling. Ponce (199) indicated that the acceleration term in the Saint Venant Equation for flood routing in natural channels can be neglected. Hence, the equation turns into the diffusion wave equation. Also, he exhibited that the form of equation includes inertia forces (Ponce, 199). Even if different forms of the Saint Venant equation (kinematic, noninertia, quasi steady dynamic and dynamic) are used, the equation can always be changed into the diffusion wave equation form (Yen and Tsai, 1). The Saint Venant equation can be also written as a two-dimensional equation in overland flow problems (Tayfur et al., 1993), and then it can be transformed into the diffusion wave equation form. The DQM was developed by Richard Bellman in The method offers a solution to the differential equations of any systems. The governing equations of the method include existing

2 136 / JOURNAL OF FLOOD ENGINEERING (JFE) initial and boundary conditions (Bellman and Casti, 1971; Bellman et al., 197). In the Faculty of Plane/Space, Oklahoma University, Bert and his team conducted studies in which DQM was used for the static and dynamic analyses (calculation for free vibration frequency) of the layered and composite plates (Bert and Malik, 1996). Using DQM, Shu and Richards (199) made the studies on some of the fields such as fluid mechanics, buckling and bending of plates and beams. In some recent studies (Fung, 3), principles of DQM were applied for the solution of the initial value problems encountered in the fields of heat transfer, and fluid mechanics. In his Ph.D. thesis, Civalek (3) examined the free and forced vibrations with single and multiple degrees of freedom using Harmonic Differential Quadrature Method (HDQM). Civalek (4) made the studies on application of DQM and HDQM for buckling analysis of thin isotropic plates and elastic columns. Shu et al., (3) presented a local radial basis function-based differential quadrature method. Shu et al., (4) examined the DQM used to simulate the eccentric Couette Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. Lo et al., (5) presented natural convection in a differentially heated cubic enclosure after solving the velocity vorticity form of the Navier Stokes equations by a Generalized Differential Quadrature Method (GDQM). Ding et al., (6) examined, the Local Multiquadric Differential Quadrature Method (LMDQM) applied on three-dimensional incompressible flow problems. Since the general form of St. Venant equations are nonlinear, Hashemi et al., (6, 7) used Incremental Differential Quadrature Method (IDQM) for the solution. When the studies are examined, it is seen that the diffusion equation approach and DQM are used in different applications. However, DQM for the numerical solution of diffusion wave equation has not been employed. In this study, St. Venant equations are transformed into the diffusion equation and DQM is applied to generate the solution.. THEORY OF THE DIFFUSION WAVE EQUATION Solutions of unsteady flow in channels, the Saint-Venant equations or their approximations; the kinematic wave, noninertia wave, gravity wave and quasi-steady dynamic wave are defined depending on the relative importance of local inertia, pressure gradient, gravity, and friction effects involved in the physical mechanisms. One-dimensional, unsteady, gradually varied open channel flows in prismatic channels can be expressed as: A Q + = q t x (1) Q ( QV) y kt + kc Vxq + kpga kfga( S Sf ) = t x x where Q is the flow discharge; y, flow depth; A, flow cross sectional area; q, lateral outflow per unit length along the channel (infiltration, seepage, precipitation, etc.); V x, x component velocity of lateral flow; S, channel bed slope; S f, friction slope; x, longitudinal coordinate; and t is the time. In Eq. the expressions multiplied by k t, k c, k p, define local acceleration, convective ()

3 DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION... / 137 acceleration, and pressure exchange, respectively. Here, k f is a coefficient that multiplies channel bed slope and friction slope. The coefficients k c, k t, k p, k f equal or 1 depending on the wave approximation considered (Yen and Tsai, 1). The wave approximations which are used for the solution of St. Venant equations are summarized in Table 1. Noninertia wave approximation is also referred to as the diffusion wave equation (Akan and Yen, 1977). However some researchers include the inertia effects in the diffusion wave model (Ponce, 199; Sivapalan et al., 1997). Both linear and nonlinear waves are diffusive and can involve the inertia effects. It is possible to linearize the St. Venant equations and solve using different wave approximations. The linearized St. Venant Equations can be converted to the diffusion wave equation for different wave approximations (Yen and Tsai, 1). In this study, the St. Venant equations written in the form of diffusion wave equation are solved by DQM. In practice, the ratio of lateral flow of rainfall, seepage, or infiltration into the flood-induced discharge is relatively small. In addition, the x-component velocities of lateral flow of rainfall, seepage, or infiltration are also negligible compared to the flow velocity. Therefore, the linearized combination form of the governing equations for unsteady and gradually varying flow in prismatic open channels can be expressed as Eq. 3 assuming Q = Q + Q and A = A + A (Yen and Tsai, 1). Q Q Q F Q 1 Q + C = m ( kp kcf ) ( k ) t + kc k t t x x u x t gyh t (3) where, Q is the uniform flow discharge; A is the uniform flow cross-sectional; Q, perturbed flow discharge; A, perturbed flow cross-sectional area; u, uniform flow velocity; and y, uniform flow depth. The coefficient m, the steady uniform flow F, steady uniform hydraulic depth y h, surface width B, and the wave celerity C are defined in Eq s. 4a-4d, respectively; uy m = h S (4a) F = u gy h (4b) A y h = B Sf / A C = u S / Q f (4c) (4d) Differentiating Eq. 3 with respect to x and similarly with respect to t, and applying the results to Eq. 3, after neglecting the third order terms, yields:

4 138 / JOURNAL OF FLOOD ENGINEERING (JFE) Q Q ( p c ) ( t c) C C C k k F k k F Q + = + + kt F m. (5) t x u u x Eq. 5 gives a mathematical description of the linearized shallow water with the assumption that high-order effects are ignored. For the dynamic wave (k t = k c = k p = k f = 1), Eq. 5 can be simplified to: Q C Q D Q + = h t x x (6) where D h is the hydraulic diffusivity coefficient: C C Dh = ( kp kcf ) + ( kt + kc) F kt F m. (7) u u The celerity used in Eq. 7 depends on the channel cross-section geometry, the area of flow, and the resistance formula (S f ). For a trapezoidal channel, C is defined as: C = α z y b zy b+ 1+ z y b+ zy u. (8) Here, α = 1 for Chezy formula, α = 4/3 for the Manning s formula, b is the bottom width, and z is the side slope. The transformation of the hydraulic diffusivity coefficient with respect to the wave approximations is given in Table 1 (Yen and Tsai, 1). Table 1 The Hydraulic Diffusivity Coefficients Depending on Wave Approximations Waveapproximations Model coefficients Hydraulic diffusivity (D h ) Dynamic wave k t = k c = k p = k f = 1 C C Dh = 1 1 F + m u u Quasi-steady dynamic wave k t = and k c = k p = k f = 1, C Dh = 1 1 F m u (1) Noninertia wave k t = k c = and k p = k f = 1 D h = m (11) Kinematic wave k t = k c = k p = and k f = 1 D h = (1) (9) These equations in Table 1 describe the generalized diffusion wave in a prismatic open channel with arbitrary cross-sectional geometry.

5 DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION... / 139 The St. Venant equations can be arranged according to y with the transform of y = y + y instead of Q: y C y D y + = h. (13) t x x 3. DIFFERENTIAL QUADRATURE METHOD (DQM) DQM is an alternative approach to the standard methods such as finite difference and finite elements, for the initial value and boundary conditions encountered in physics and mathematics (Shu and Richards, 199; Shu and Chew, 1997). In the Differantial Quadrature Method, a partial derivative of a function with respect to a space variable at a discrete point is approximated as a weighted linear sum of the function values at all discrete points in the region of that variable. The method can be written as (Civalek, 3): u u x A u x i N (14) r N ( r) x( i) = = ij ( j) = 1,,..., r x x = xi j = 1 where x j are the discrete points in the variable, u (x ) are the function values at these points and j A (r) are the weight coefficients for the r th order derivative of the function. ij Determining the weight coefficients is the most crucial step in use of DQM. The weight coefficients change depending on the u (x) function. According to the chosen u (x) function, the method can take different names like Polynomial Differential Quadrature (PDQ), Fourier Expansion Base Differential Quadrature (FDQ), Harmonic Differential Quadrature (HDQ) (Shu et al., ). DQM performance is highly dependent on the boundary conditions and sampling grid points. The overall sensitivity of the model especially depends on the location and number sampling grid points. Civalek (3) implies that the determination of the effective choice for any problem reduces the analysis time. For instance, previous studies show that, in the problems that have linear equations and homogeneous boundary conditions, selecting equal intervals are adequate. In vibration problems, the choice of grid points through the Chebyshev-Gauss-Lobatto method is more reasonable. In time-bound equations and in initial value problems, selection of unequal intervals for sampling grid points produces the appropriate solutions (Civalek, 3). 4. DQM FOR THE DIFFUSION WAVE EQUATION The general convective-diffusion equation as seen in Eq. 15, can be rewritten for the solution of DQM as seen in Eq. 16 t x x Q + C Q = D Q h (15)

6 14 / JOURNAL OF FLOOD ENGINEERING (JFE) A Q + C B D B Q = i= 1,,..., N; s= 1,,..., R R N N () rs, ir, ji, h ji, js, r = 1 j = 1 j = 1 (16) where N, is the number of sampling grid points in the x direction; R is the number of calculation points in time axis; and A, B, B () are the weight matrix coefficients. Determining the boundary conditions, Eq. 16 is solved for the Q (i, s) values. For example, when Q (x,) = f (t) and Q (, t) = f (t), Eq. 16 yields 1 R N N () Ars, Qir, + C Bji, Dh Bji, Qj, s= A1, sqi,1 ( cb1, i DhB1, i) Q1, s r = j = j = (17) where, i =, 3,..., N and s =, 3,..., R. Using the properties of Legendre polynomials, the weight coefficient matrix can be written as (Shu et al., 4): L i = N ( tj ti) (18) j = 1 B i, k = Li L ( x x ) k i k i k (19a) N B i, i = B, i k (19b) k = 1 ik B B () = () i, k i, k Bi, kbi, k xi xk i k (a) N B () = () B i, i ik, k = 1 i k (b) For the numerical discretizatizon, several different methods are available. It can be selected with equal intervals, or non-equal intervals like Chebyshev-Gauss-Lobatto grid points, or with the normalization of the routes of Legendre polynomials (Shu, ). In the present study, several different approximations have been tested, and, Chebyshev-Gaus-Lobatto grid points are selected. In the Chebyshev-Gaus-Lobatto approximation, by writing: 1 i 1 ri = 1 cos π N 1. (1)

7 DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION... / 141 The locations as the calculation points can be calculated as follows: x i = ri r1 r r N 1. () In time domain, these: yields ri r1 t i =. (3) rr r1 By considering R, in place of N, in Eq. 1 and Eq.. A matrix can be written like B matrix as follows: L i = R ( tj ti) (4) j = 1 A i, k = Li L ( t t ) k i k i k (5a) R A i, i = A, i k (5b) k = 1 5. APPLICATION 5.1. Numerical examples 1 The example solved was taken from the study of Bajracharya and Barry (1997). The following entrance boundary condition was used to test the optimality results presented above: t Q(, t) = t exp 1. (6) The initial condition Q (x, ) = Q = was assumed. The values of C and D were taken as i h 1 m/s and 1 m /s, respectively. For natural streams, average C and D h values are predicted as presented in the Flood Studies Report III (1975) (Bajracharya and Barry, 1997). The DQM method results at t = 5 sec. are shown in Fig. 1, where N x is the number of grid points in x direction, N t is the number of grid points in t direction. And, Explicit Finite Difference Method (EFDM) and Implicit Finite Difference method (IFDM) results are shown in Figs and 3, respectively. The comparison of DQM, Explicit Finite Difference Method (EFDM) and Implicit Finite Difference method (IFDM) results are shown in Fig. 4. According to the results taken from the three methods, it is obvious that EFDM and IFDM results are highly affected by the number of grid points. However, the DQM method results were converged rapidly using a small number of grid points. This feature of DQM has also been verified by Tezer-Sezgin (4) and Kaya (9). ik

8 14 / JOURNAL OF FLOOD ENGINEERING (JFE) 3 Q(m 3 /sec) Nx - Nt X(m) Figure 1: DQM Solutions According to the Different Number of Grid Points 3 Q(m 3 /se c) Nx-Nt X(m) Figure : EFDM Solutions According to the Different Number of Grid Points

9 DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION... / Q(m 3 /s ec) Nx-Nt X(m) Figure 3: IFDM Solutions According to the Different Number of Grid Points 35 Q(m 3 /sec) Method (Nx-Nt) DQM (4-51) EFDM(1-51) IFDM(1-51) DQM (1-15) EFDM(11-11) IFDM(11-11) X(m) Figure 4: The Results Obtained from DQM, EFDM and IFDM

10 144 / JOURNAL OF FLOOD ENGINEERING (JFE) In the EFDM and IFDM, as the number of grid point s increases, the finite difference results converge the DQM results. However, increasing the number of the grid points causes the extensive process time. As seen in the Figures, the results taken with 1 15 (N x N t ) grid points are not so different from the results taken with 4 51 grid points. The results taken with 1 15 grid points can be taken with 1 51 grid points in EFDM, and 1 51 grid points in IFMD. In this condition the solution times are.14, 4.44 and 14.6 seconds in the DQM, EFDM, IFDM models, respectively. Furthermore, the stability of the results in EFDM and IFDM can be disrupted according to theand values chosen. For the stability of the solution, theandvalues must be chosen appropriately; nevertheless, in literature relating DQM, there is not any criteria for choosing theandvalues. This study provides some practical values for a proper selection. For instance, 1 through 45 is a suitable range; out of this range is not convenient, subsequently resulting inaccurate result or excessive computing time, vise versa. A previous study (Kaya, 9) on DQM has demonstrated that the crucial number for the optimum value of the solution points is about 8-1; using lower than this interval considerably increases the error based on the analytical solution. 5.. Numerical Examples Flood propagation between Gundagai and Wagga Wagga stations over the Murumbidgee River in New South Wales, Australia (Sivapalan et al., 1997) was examined as a second numerical example. Gundagai, which is located upstream of the river with 1 km length and.3% longitudinal slope, measurements were undertaken as initial data, and then flood hydrograph Figure 5: DQM Results, Sivapalan et al., (1997) and Actual Measurements at Wagga Wagga

11 DIFFERENTIAL QUADRATURE METHOD FOR NUMERICAL SOLUTION... / 145 Figure 6: Relationship of Number of Sampling Points to Discharge for DQM at Wagga Wagga was determined. A comparison was conducted taking into account DQM solution for constants C and D h values, actual measurements and FDM solution performed Sivapalan et al. (1997). The DQM provided the better results than FDM even though using the less sampling points. The results of such computations are presented in Fig. 5. In addition, Fig. 6 gives how the results changes associated with the number of sampling points. 6. CONCLUSION The St. Venant equations under some acceptance can be translated to linear form, and than can be solved by numerical solution methods. In this study, diffusion wave approach and differential quadrature methods were employed for one-dimensional, gradually varied, unsteady openchannel flow. The results were compared with the results using finite difference methods. In contrast to FDM, a small number of grid points are sufficient for a stable solution using DQM and there is no stability problem associated with the number of grid points and the time interval. Furthermore, the number of grid points and time interval can be selected independently since they do not interfere with each other. Less sensitivity to the number of grid points is an advantage of DQM, so the model provides thw reasonable results in a short time even though it uses fewer nodes and consequently less memory usage. In FDM s, similar results with DQM could be obtained, only when the number of grid points was increased. Based on the comparison of their best solutions, it seems that the DQM provides solutions much faster than FDM s. References [1] Akan A. O., and B. C. Yen, A Nonlinear Diffusion Wave Model for Unsteady Open Channel Flow, Proceedings of the 17 th IAHR Congress, August, Baden-Baden, Germany,, (1997), [] Appleby F. V., Runoff Dynamics: A Heat Conduction Analogue of Storage Flow in Channel Networks, International Association of Scientific Hydrology, Assemblee Generale de Rome, 38(3), (1954),

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