Numerical Model And Software For Simulation And .

6th Int'l Conference on Advances in Engineering Sciences and Applied Mathematics (ICAESAM’2016) Dec. 21-22, 2016 Kuala Lumpur (Malaysia)Numerical Model and Software for Simulationand Visualization of Flooding in the Urban AreaSaeful Bahtiar1, Somporn Chuai-Aree1, Anurak Busamun1, and Areena Hazanee1 Abstract—The shallow water equation system was applied in thisstudy to model the water flow over complex topography in the urbanarea. A finite volume method based on the first order well-balancedscheme was used to solve the model. In order to upgrade the capabilityof flood control in urban areas, we developed in this work thenumerical software for simulation and visualization of flooding in theurban areas. The developed software was used for the flood simulationin Jakarta, Indonesia. The simulation results can show effects ofbuilding for the urban flooding.Keywords— Shallow water equations, Finite Volume Method,Simulation, Visualization.I. INTRODUCTIONLOOD is one of the catastrophic events that can result toseveral effects such as loss of human life and loss ofeconomics. Flood is the most common natural disasterwhich often occurs in almost countries all over the world.Flooding may be caused by several reasons, such as heavy rains,dam-break, flood embankments [10]. Previous studies haveshown that the use of mathematical model can simulateflooding in urban areas, for example: [10], [9], and [3].The shallow water equations (SWEs) are one of themathematical models which describe the surface flow overcomplex topography and also used to simulate manyphenomena of practical interest including river flood, tsunamipropagation and dam break flow [5], [1], [13]. Solutions ofshallow water equations are difficult to be solved analytically,and numerical methods are needed.There are many approaches of numerical methods that arewidely used for solving the SWEs. They include the finiteelement method (FEM), the finite volume method (FVM) andthe finite different method (FDM), etc. The most of researchersselected to use the finite volume method for the water flowsimulation. For the finite volume method, [8] developed arobust, accurate and computationally efficient numerical modelto solve shallow water equations, and finite volume methodbased on triangular grid is used, while [2] studied the finitevolume method to solve shallow water equations and Godunov-FSaeful Bahtiar1 is with the Mathematics and Computer Science Department,Prince of Songkla University, Pattani, Thailand .Somporn Chuai-Aree1, Anurak Busamun1 and Areena Hazanee1 are with theMathematics and Computer Science Department, Prince of Songkla University,Pattani, pe method based on the approximate Riemann solver is usedto compute the flux function.For the numerical methods, the solutions obtained are alwaysrelated to computer programing because numerical calculationsare often repeated every time to calculate the solutions.Computer simulation based on software development isnormally used to minimize fatalities and damage to publicservice facility for disaster preparation and prevention process.Computer programs were developed in previous works forflooding simulation. [4] Developed lizard software forsimulation of flooding, while [1] simulated and visualizedrainwater flow in the surface by developing the software.Moreover, numerical techniques were studied for improvingthe numerical model. [12], [11] and [7], provided twoimportant factors for solving shallow water equations that aresteady-state stationary and contact discontinuities. Both thecompletions are not always easy to be handled numericallyusing standard numerical schemes. This difficulty can beovercome by using the well-balanced schemes [7]. [12] used asecond order scheme based on linear reconstruction for solvingwell-balanced scheme and prove the scheme can preserve thenon-negativity water height. [11] Develop a first order schemewith hydrostatic reconstruction to solve well-balanced scheme.[6] proved that the first order based on hydrostaticreconstruction can preserve the computation in the wet-dryfront area.In order to upgrade the capability of flood control in theurban areas, we developed in this work a numerical model andsoftware for simulation and visualization of flooding in theurban area. The FVM based on the first order well-balancedscheme is used to solve the shallow water equations.The paper is organized as follows. Section 2 presents shallowwater equations, while section 3 provides the details about thenumerical method. Section 4, the application of city floodsimulations with building and without building is shown. Theresults in application are given in section 5, and conclusion ispresented in section 6.II. SHALLOW WATER EQUATIONSThe model in this work was developed based on shallowwater equations for determining the behavior of water flow inthe urban area. The model system is presented in vector form asfollow:

6th Int'l Conference on Advances in Engineering Sciences and Applied Mathematics (ICAESAM’2016) Dec. 21-22, 2016 Kuala Lumpur (Malaysia) q f q g q z q s q t x y(1) where q h uh vh T is the vector of dependent variablesconsisting of the water depth h, the discharge per unit width uhand vh with velocity component u and v in the x and ydirections, and t is the time. The vector f and g can be expressed in terms of theprimary variables u, v and h as uh vh 2 1 2f q u h gh , g q uvh , 2 21 2 uvh vhgh 2 . (2) R 0 S q S fx , Z q ghS 0 x ghS S0y fy where f q and g q are the flux vectors of the system inthe directions of the coordinate axis x and y, respectively. g isthe acceleration due to gravity, R is the rate of river increasedby rainfall. S 0 x and S 0 y are the bed slope in x and ydirections, respectively. S fx and S fy are the friction terms in xand y directions, can be estimated by using the Manningresistance law.S fx gn2u v uh2 h 432, S fy gn2u v vh2Fig. 1 The notation used for a Cartesian 2D grid (a) and typicalstructured grids (b)Integrating the conservation law in (1) over each grid cell,and by applying Gauss divergent theorem to the second andthird terns, we obtain: Ci , j qdxdy F n d t Ci , j(4)Ci , jthecell boundary.Dividingwiththecellarea i, j xi, j y i, j , and denoting Q i , j , Z i , j , and S i , j as the average of q , z q and s q over the cell C i , j respectively,the (4) becomes, Qi , j t(3) where C i , j is the control volume, is the boundary of the C i , j . F f q , g q is the flux vectors at each interface of2 h 4 3 z q dxdy s q dxdy 1 i, j F n d Z i , j S i , j(5)where subscripts i, j denote spatial index of the cell.The line integral in (4) can be calculated by the sum of thefluxes over the four walls around the cell as shown in Fig. 1 (b).By using the first order well-balanced scheme in [11], theIn which n is the Manning resistance coefficient.III. NUMERICAL METHODgravity force Z i , j can be distributed to the numerical fluxes forThe finite volume methods based on the first order wellbalanced scheme is used to solve the model. The details can beshown as the followings.each sub-interface, and Euler's method can be used forapproximation of the time derivative, so we can write the finitevolume formulation asA. Finite Volume Discretization Over StructuredFor the finite volume method, the numerical domain issubdivided into rectangular (interval) grid cells of the form t * t * Qin, j 1 Qin, j f i 1 , j f i* 1 , j g i , j 1 g i*, j 1 tSi , j (6)2222 x y * *where f i 1 , j and g i , j 1 are the numerical fluxes function Ci , j x 1 , x 1 y 1 , y 1 as show in Fig. 1(a). Let i 2 i 2 j 2 j 2 x x 1 x 1 and y y 1 y 1 are the length of thei 2i 2j 2j 2 2 2ndepending upon the chosen scheme. Qi , j represents the cellaverage value over the (i,j) grid cell at time tn.B. Interface Fluxes Calculationcells.The fluxes calculation based on the first order well-balancedscheme was used in this work. The fluxes with gravity force ateach interface are computed based on the Harten, Lax and VanLeer (HLL) Riemann solver with good robustness and accuracyhttps://doi.org/10.15242/IIE.E121603076

6th Int'l Conference on Advances in Engineering Sciences and Applied Mathematics (ICAESAM’2016) Dec. 21-22, 2016 Kuala Lumpur (Malaysia)as showed in [9]. The application of this approach to thenumerical fluxes can be expressed as: (7)f i* 1 , j f HLL U i 1 , j , U i 1 , j Lix 1 , j2222 g i* 1 , j g HLL U i , j 1 , U i , j 1 Liy, j 1222 2(8) Where Lix 1 , j and L y 1 are well-balanced, can bei, j 22expression based on work [11] with 0 x g ˆ2 Li 1 , j hi 1 , j 222 0 where f HLL and 0 y , L 1 0i, j 2 g ˆ2 hi , j 1 2 2 C. Source Terms ComputationThe source term vectors S i , j in (2) consists of the rate ofn 1(9)(6), ashin, j 1 hin, j tRin, j nin each cell (i ,j) in the range of time step for n to n 1. For thesecond one, the friction forces can be computed by the semiimplicit method as showed in [1]. It update a new value ofuhin, j 1 in (6) as follows:uhin, j 1 guarantee that water depth is nonnegative. The formulas for ĥare given by (14)S L min u L ghˆ L , u R ghˆ R , 0 (15)SU max v D ghˆ D , vU ghˆU , 0 (16) min v D ghˆ D , vU ghˆU , 0 (17)SD(19) u v h n 2i, jn 2i, jn 1 4 / 3i ,, jgetvh in, j 1vh in, j 1 1 tgn2(20) h 2u in, j2v in, jn 1 4 / 3i ,, jThe semi implicit method allows to preserve stability andsteady state at rest.D.S R max u L ghˆ L , u R ghˆR , 0 gn 2Similarly, vh in, j 1 can be compute using formula so we can max 0, hi , j z i , j max z i , j , z i , j 1 the work of [14] as follows:uhin, j 11 thˆi 1 , j max 0, hi , j z i , j max z i , j , z i 1, j (12) ˆhi , j 1 2 In order to completely determine the numerical flux in HLLRiemann solver, there is need to estimate the wave speedsS R , S L , SU and S D . The wave speeds are assigned based on(18)Where Ri , j is the average rate of river increased by rainfall,Audusse’s scheme for preserving the lake-at-rest condition and2gives y xv ghˆ and v ghˆ .They are computed with the hydrostatic reconstruction state Tby U hˆ uhˆ vhˆ when ĥ is a particular value based on and for g q f qeigenvalues of Jacobian matrixcontrol volume per unit time, we update a new value of hi , j in g HLL are approximate Riemann solver and u ghˆ are the smallest and largestriver increased by rainfall and the friction force, For the firstone, because this one is water depth that is vertically added tobased on Harten, Lax and Van Leer (HLL) and can be definedby HLL f U L S R f U R S L S L S R U R U LfU L ,U R (10) S R S L g U D SU g U U S D S D SU U U U D HLL (11)gU D ,UU SU S D When u ghˆStability ConditionThe Courant Friedrich Lewy (CFL) condition, which is thestability criterion of explicit numerical schemes is used todefine the time step t where t 0.5 Amin(21) max In the (21) Amin min xi , j , yi , j , for all (i,j)where D is the computational domain, while max D,is themaximum absolute value of all the wave speeds in thehttps://doi.org/10.15242/IIE.E121603077