Radial Basis Functions Alternative Solutions to ShallowWater Equations

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Radial Basis Functions Alternative Solutions to Shallow Water Equations

Y. Alhuri¹, D. Ouazar²and A. Taik³

1,3Dept of Mathematics, UFR-MASI, FST, HassanII University, Mohammedia, Morocco

2 Dept of Genie Civil, LASH, EMI, Mohammed V University, Rabat, Morocco

1 Corresponding Author E-mail: [email protected]

Abstract. In this paper, shallow water equations (SWE) are solved through a variety of meshless meth- ods known as radial basis functions (RBF) methods. RBF based Meshless methods have gained much attention in recent years for both the mathematics as well as the engineering community. They have been extensively popularized owing to their flexibility, power and simplicity in solving partial differ- ential equations. The technique is of quasi-analytical type and based on the collocation formulation and does not require the generation of a grid or integrals evaluation. A bundle of techniques such as MQRBF, IMQRBF, GSSRBF, CSRBF are formulated for the specific case of shallow water equations (SWE), with adequate parameters, these techniques show robustness and low computational costs. For validation purposes, two applications are presented. One deals with Burger’s equation (2- D), and the second with a linear two-dimensional hydrodynamics model of rectangular channel shape. Finally a third application concerning a real case study of the hydrodynamic of the Ourika valley in Morocco is investigated aiming at rebuilding the 1995 historical flood wave propagation, therefore delineate inundated areas, estimate velocities and wave time arrivals.

Key words: Water resources, Burger’s, Shallow-water equations, Radial Basis Functions (RBFs), Floods, Ourika valley.

1. Introduction

Water resources cover a wide range of problems including preservation of water supply, watershed management, dam construction for mitigating ﬂoods and/or conservation purposes, river, estuaries and lake management and restoration, groundwater, pollution control, to cite a few examples.

Due to inherent complexity originating from the turbulent nature of flows, the specificity of the domain geometry which can be very often natural (river, lake, lagoon, estuary, watershed, aquifer, etc..) or man built and/or regularized (artificial canals, intakes, etc..), historical nature of some data sets and some other special effects (soil roughness, presence of air, wind, etc), it leads to many difficulties associated to the identification of domain boundaries, boundary conditions, and grid generation in addition to the already complex governing equations.

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The governing equations are derived from the conventional conservation laws of mass, momentum, and energy, and therefore are very similar to those of fluid mechanics. Navier-Stockes (NS) and Reynolds averaged Navier-Stockes (RANS) variants (depth averaged, turbulence closure models, friction factors, simplifications) are the governing equations for many flow situations.

In this paper we will restrict ourselves to the so-called shallow water equations (SWE) that are derived by depth averaging of the RANS equations, reducing the dimensionality of the problem by one and thus computational time. But these equations require discretisation when using conventional numerical techniques such as finite difference, finite element or finite volume methods leading to high computational costs.

After an introduction to one of the basic fluid mechanics equations namely Burger’s widely known to represent a class of physical problems and to serve for numerical benchmarks, both linear and non linear wave propagation problems are considered. They are widely applied for floods / dam breaking, weirs, tidal flows and storm surges.

The computational cost of discretisation based methods is avoided by the use of a mesh-free method known as radial basis functions (RBFs) method. Applications of radial basis functions have gained quite some importance over the past years. They have been successfully applied to large variety of problems [4, 7, 8, 9], [11, 12, 13, 15] and [2, 20, 26]. A major advantage of using RBFs is that the collocation points on the domain do not need to be uniformly spaced. One of the most popular choices for global RBFs is the multiquadric (MQ) approximation first introduced by Hardy [19]. In Kansa’s method [17], a function is first approximated by an RBF, and its derivatives are then obtained by differentiating the RBF. Since (MQ) is infinitely smooth function [18], The corresponding numerical results demonstrated that the MQ scheme was more accurate and efficient than the other traditional methods. The MQ method, however requires one to solve a system of equations with a full matrix. This can be extremely expensive when there are hundreds of collocation points.

The construction of compactly supported radial basis functions (CSRBFs) by Schaback, Wendland and Wu in [23, 24, 28] has made it possible to overcome the non sparsity of the matrices arising from global RBFs techniques because CSRBFs can convert the global scheme into a local one. Furthermore, radial functions of compact support are now being proposed leading to positive deﬁnite matrices and genuinely banded interpolation matrices. In [25], Wendland gives a general formulation, which in- cludes the Buhmann’s ones [5]. But the proper choice of the support radius and the smoothness of such functions require some skills and experience of the end user.

In this paper, we set-up efficient algorithms based on Kansa’s RBF and Wendland’s CSRBF for simulating large scale time-dependent problems for SWE. The RBFs are employed for spatial approximation in conjunction with an explicit forward difference scheme in time. The proposed scheme is first applied to classical Burger’s equation in 2-D, and to linear SWE in a rectangular channel for validation purposes. The RBF scheme is then applied to non-linear SWE, to simulate water elevations, velocities and wave time arrivals in a real case study ( Ourika Valley in Morocco) to rebuild the 17th August 1995 historical flood effects and delineate inundated areas.

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2. RBF background

2.1. Globally supported radial basis functions

The principal idea of the radial basis interpolation is to interpolate a ﬁnite series of an unknown function f(X)atN distinct pointsXj onΩby the following expansion

f(X) N

j=1

αjϕ(X-X^j), (2.1)

where αj’s are the unknown coefﬁcients to be calculated at each time step, and ϕ(X-X^j) is the radial basis function, Xj ∈ Rⁿ, j = 1,2, ..., N, X-Xj = rj, rj is the Euclidean distance rj = (x−xj)²+ (y−yj)² and X = (x, y), X^j = (xj, yj). The most popular RBF is Multiquadric (MQRBF) (r_j² +c²)^1/2, Inverse Multiquadric (IMQRBF) 1/

(r²_j +c²)^1/2, and Gaussian (GSSRBF) e⁻^(r/c)² where c = 0 is the shape parameter controlling the ﬁtting of a smooth surface to the data.

The lack of the mathematical theory makes it very difﬁcult to choose a suitablecfor the current study.

Kansa in [16, 17] found that the computational accuracy of the interpolation can be improved by vary- ing the shape parameter with the basis function, he employed numerical experiments to resolve the optimal shape factorc² given byc²_i =c²_min

c²_max/c²_min(i−1)/(N−1)

,wherec²_minandc²_max are two input parameters. For more details see [10]. The analysis of Wong et al. [26] showed that a near-optimal approximation of the hydrodynamic model can be achieved by using the proposed value 0.815dmin, whered_minis the minimum distance between any two adjacent collocation points andhis the average water depth between the two points.

Micchelli [21] has proved that the system of equations (2.1) and (??) has a unique solution for distinct collocation points. In accordance with this fact Powell (1992) [22] showed that the interpolation matrix of the MQRBFs is non-singular for any set of distinct points, and that the invertibility is guar- anteed. Therefore adding polynomial terms to the MQRBFs interpolation is not generally required.

However, this argument may not be applicable to other RBFs [27].

2.2. Locally supported radial basis functions

In numerical analysis, the concept of locally supported basis functions is of general importance. Sev- eral functions spaces used for approximation possess locally supported basis functions as compactly supported radial basis function (CSRBFs). The general advantages of compactly supported radial basis functions are a sparse interpolation matrix on the one hand, and the possibility of a fast evaluation of the interpolate on the other hand. Thus, it seems to be natural to look for locally supported functions also in the context of radial basis function interpolation [25].

A family of CSRBFs {ϕl,k} was first introduced by Wu in [28] and later expanded by Wendland [24] in the mid1990s. CSRBFs must be strictly positive definite in R^dfor all dless than or equal to fixed valued0. Generally a CSRBFϕl,k(r)is expressed in the form

ϕ_l,k(r) = (1−r)ⁿ₊p(r), for k ≥1, (2.2) with the following condition

(1−r)+ =sup{0,(1−r)}, (2.3)

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where p(r) is a prescribed polynomial, r is the Euclidean distance. The index l is the dimension number and 2k is the smoothness of the function. The basis function ϕl,k(r) can be scaled to have compact support on [0, δ]by replacing r with r

δ forδ > 0. Again, the radial basis interpolation is to interpolate a ﬁnite series of an unknown function f(x, y)at N distinct points on Ωby the following expansion

f(x, y) N

j=1

αjϕ^δ_l,krj

δ_j

, (2.4)

where ϕ^δ_l,k is the compactly supported radial basis function, α_j’s are the unknown coefﬁcients to be calculated at each time step and the scaling factorδj can be variable or constant at different node points j = 1,2, ..., N depending on the nature of the problem [26].

3. Numerical results

3.1. 2-D Burger’s equations

In this example we solve the nonlinear Burger’s model, which has widely been used in ﬂuid dynamics as a simpliﬁed model for turbulence, boundary layer behavior, shock wave formation and mass transport.

The equations to be solved are given by :

∂u

∂t = α∆u−(uux+uuy), (x, y)∈Ω⊆R²,0≤t≤T, (3.1)

u = g(x, y, t), (x, y)∈∂Ω, (3.2)

u = h(x, y,0), (x, y)∈Ω, (3.3)

with ∆u = uxx +uyy in 2D, such thatux = ∂u

∂x, uy = ∂u

∂y, ux = ∂²u

∂x², uy = ∂²u

∂y² . The functions g(x, y, t)andh(x, y)are chosen accordingly so that the exact solution is :

u(x, y, t) = 1 1 +exp

x+y−t 2α

. (3.4)

This problem was also considered in [6, 14]. For time integration, we use of the implicitΘ-weighted scheme. Therefore equations (3.1) will be of the form :

u(x, t+ ∆t)−u(x, t)

∆t = Θ(α∆u(x, t+ ∆t) +f(x, y, t+ ∆t))

+(1−Θ)(α∆u(x, t) +f(x, y, t)), (3.5) which leads to :

uⁿ⁺¹

∆t −Θ(α∆uⁿ⁺¹−uuⁿ⁺¹_x −uuⁿ⁺¹_y ) = uⁿ

∆t + (1−Θ)(α∆uⁿ−uuⁿ_x −uuⁿ_y), (3.6) we study numerically equations (3.6) by using Wendland’s compactly supported radial basis function CSRBF interpolation and Kansa’s multiquadric radial basis function MQRBF. We generated400 collocation points;320collocation points are in the interior and80collocation points are on the boundary.

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At each time stepn, the values of the variablesuⁿat the interior points are calculated using this equation, while at the boundary points the variablesuⁿ are determined by the given boundary conditions.

This problem was computed for the case : α = 0.05, 0 ≤ t ≤ 1.25with compact support δ = 10, shape parameterc= 0.2, time step∆t = 0.01s,Θ = 0.5, Kansa’s Multiquadricφ(r) =√

r²+c² and Wendland CSRBFφ3,2

r δ

= (1−(r/δ))⁶₊(35(r/δ)²+ 18(r/δ) + 3).

The numerical solution is evaluated at scattered collocation points and the spatial partial derivatives are formed directly from partial derivatives of the radial basis functions without using any difference scheme. Figures (1-4) illustrate respectively the numerical results in different times and compare the simulation of (MQRBF) and (CSRBF) with the exact solutions. The present method is easily imple- mented, stable, effective and produces highly accurate numerical results comparable to the analytical solution and to Method of Fundamental Solution MFS - Dual Reciprocity Method DRM which is given in [14]

0 0.5

1 0

0.5 0 1

0.5 1

y MQ R B F

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.5

1 0

0.5 0 1

0.5 1

y C S R B F

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.5

1 0

0.5 0 1

0.5 1

y Ana lytica l

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1: Approximation solutions by (MQRBF, CSRBF and Analytical ) att = 0.5swithα = 0.05, c= 0.2,δ= 10,∆t= 0.01s.

0 0.5

1 0 0.5

1 0

0.02 0.04

y E rorr B y MQR B F t =0.5s

x

Error

0.005 0.01 0.015 0.02 0.025

0 0.5

1 0 0.5 0 1

0.02 0.04

y E rorr B y C S R B F t =0.5s

x

Error

0.005 0.01 0.015 0.02 0.025

Figure 2: Errors between MQRBF, CSRBF solutions and Analytical one att = 0.5s withα = 0.05, c= 0.2,δ= 10,∆t= 0.01s.

4. Shallow water equations

4.1. Linear shallow water equations

We will compare different types of RBFs for shallow water equations (SWE) and study the conver- gence of these RBFs schemes. In order to compare them with an analytic solution, the linear SWE is used for a simple sensitivity analysis. This linear SWE is obtained by neglecting the wind stress,

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0 0.5

1 0 0.5 1

0 0.5 1

y MQ R B F

x 0

0.2 0.4 0.6 0.8 1

0 0.5

1 0 0.5 1

0 0.5 1

y C S R B F

x

0 0.5

1 0 0.5 1

0 0.5 1

y Ana lytica l

x 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Figure 3: Approximation solutions by (MQRBF, CSRBF and Analytical ) at t=1.27s with α = 0.05, c= 0.2,δ= 10,∆t= 0.01s.

0 0.5

1 0 0.5 0 1

0. 02 0. 04

y E rorr B y MQ R B F t =1. 27s

x

Erro r

0. 005 0. 01 0. 015 0. 02

0 0.5

1 0

0.5 1

0 0. 02 0. 04

y E rorr B y C S R B F t =1. 27s

x

Erro r

0. 005 0. 01 0. 015 0. 02

Figure 4: Errors between MQRBF, CSRBF solutions and Analytical one at t = 1.27swithα = 0.05, c= 0.2,δ= 10,∆t= 0.01s.

bottom friction, diffusion and the coriolis force terms [20]. The numerical results obtained by global Multiquadric (MQRBF), Inverse Multiquadric (IMQRBF), Gaussian (GSSRBF) and local Compactly Supported (CSRBF) are compared with analytical solution. The equations to be solved are given by

∂η

∂t +H(∂u

∂x +∂v

∂y) = 0, ∂u

∂t +g∂η

∂x = 0, (4.1)

∂v

∂t +g∂η

∂y = 0, (4.2)

the analytical solution to this problem is known and given by [26]

η(x, y, t) =η0cos ω

√gH(L−x) cosωt

cos(^√^ω_gHL), (4.3)

u(x, y, t) =−η₀ g

Hsin ω

√gH(L−x) sinωt

cos(^√^ω_gHL), (4.4)

v(x, y, t) = 0, (4.5)

whereηis the water surface elevation,u,vare the depth-averaged advective velocities inx, ydirections respectively. The MQRBF and CSRBF algorithm are applied to a rectangular channel with length L = 872km, widthW = 50kmand mean water depthz_f = 20m, such thatH = z_f +η. The initial and boundary conditions are taken to correspond to the analytical solution. For the time integration scheme, we used the Euler method of second order (see [26]) :

ηⁿ⁺¹ =ηⁿ−∆tH{∇.V}ⁿ+ (∆t)²

2 Hg{∇².η}ⁿ, (4.6)

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Vⁿ⁺¹ =Vⁿ−∆tH{∇η}ⁿ+ (∆t)²

2 Hg{∇².V}ⁿ, (4.7)

where V is a vector of the depth-averaged advective velocities in x, y directions respectively. We approximate all variables and their spatial derivatives by RBFs interpolation, we generate N = 105 ,N = 205 and N = 405 collocation points in a rectangular channel. We focus on the case where N = nx ×ny = 61×5 = 405 with : 234 collocation points in the interior, 5 collocation points on the water boundary and 166 on the land boundary (see Figure 5). At each time stepn, the values of the variablesηⁿ,uⁿandvⁿat the interior points(x_i, y_i), i= 1,2, ...,234are calculated by RBFs, while the values of variables at the boundary points(xi, yi), i= 235,236, ...,405are determined by the given boundary conditions. In the computation, the excitation wave period is taken to be27hours. The values

0 1 2 3 4 5 6 7 8

x 10⁵ 0

1 2 3 4 5x 10⁴

Figure 5: Domain geometry L=872km ,W=50km, depth=20m with collocation points.

ofηⁿ,uⁿandvⁿare given by the following MQRBF radial basis function

(ηⁿ, uⁿ, vⁿ) =^N

j=1

αⁿ_j(t)

r²_ij+c², N j=1

β_jⁿ(t)

r²_ij +c², N

j=1

λⁿ_j(t)

r_ij² +c²

. (4.8) The same for IMQRBF or GSSRBF and by the following Compact Supported radial basis function (CSRBF)

ηⁿ(xi, yi, t) = N

j=1

αⁿ_j(t) 1− rij

δ 6

+

3 + 18rij

δ + 35rij

δ 2

, (4.9)

uⁿ(x_i, y_i, t) = N

j=1

β_jⁿ(t) 1− rij

δ 6

+

3 + 18rij

δ + 35rij

δ 2

, (4.10)

vⁿ(xi, yi, t) = N

j=1

λⁿ_j(t) 1− rij

δ 6

+

3 + 18rij

δ + 35rij

δ 2

, (4.11)

which are collocating with a set of data(xi, yi)^N_i=1over the domainΩwhererij = ((xi −xj)²+ (yi− yj)²)^1/2is the in 2-D. Then we will take :

Φⁿ(xi, yi, t) = N

j=1

αⁿ_j(t)ϕ(rij), (4.12)

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whereΦ = [η, u, v]^T,ϕ(r)is a radial basis function (MQ, IMQ, GSS or CSRBF). The numerical values of the unknown spatial derivatives ofΦⁿ(xi, yi)is approximated usingRBF as (herem = 1,2) :

∂^mΦⁿ

∂x^m (xi, yi, t) = N

j=1

αⁿ_j(t)∂^mϕ^{RBF s}_j

∂x^m (xi, yi),

∂^mΦⁿ

∂y^m (xi, yi, t) = N

j=1

αⁿ_j(t)∂^mϕ^{RBF s}_j

∂y^m (xi, yi).

At each time step n, the numerical solution of the vector Φ(xi, yi, t)^b+d_i=b+1 at the interior points are computed by substituting the Φ^{RBF s} and t their spatial derivatives into the system (4.1−4.2). The boundary valuesΦ(xi, yi, t)^b_i=1 are given by boundary conditions.

Figure (6) plots the simulation elevation water surface and velocity in the center point of the domain by 2D shallow water equations at 50hours. Figures (7), (8) show the simulation results of velocity distribution and3Dwater surface at50hours.

0 10 20 30 40 50 60

0. 25 0. 2 0. 15

0. 1 0. 05

0 0. 05 0.1 0. 15 0.2

T idel eleva tion a t the center point of the doma in

T ime (hrs )

Elevation x (m )

0 10 20 30 40 50 60

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

V elocity a t the center point of the doma in

Velocity u (m)

T ime (hrs ) E X AC T

MQ C S

E X AC T MQ C S

Figure 6: Analytical and computed water surface elevations and velocity in the center point of the domain,c= 0.815∗dmin,δ= 10⁷and∆t= 100sat 50 hours.

0 5 10

x 10⁵ 1

0 1 2 3 4 5

6x 10⁴ E xa ct

0 5 10

x 10⁵ 1

0 1 2 3 4 5

6x 10⁴ MQ

0 5 10

x 10⁵ 1

0 1 2 3 4 5

6x 10⁴ C S

Figure 7: Velocity distribution at 50 hours.

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0 5

x 1 0⁵ 1 0 0

2 4

6 8

1 0

x 1 0⁴ 1

0 . 8 0 . 6 0 . 4 0 . 2 0 0 .2 0 .4 0 .6 0 .8 1

E xa c t

0 5

x 1 0⁵ 1 0 0

2 4

6 8

1 0

x 1 0⁴ 1

0 .8 0 .6 0 .4 0 .2 0 0 .2 0 .4 0 .6 0 .8 1

MQ

0 5

x 1 0⁵ 1 0 0

2 4

6 8

1 0

x 1 0⁴ 1

0 .8 0 .6 0 .4 0 .2 0 0 .2 0 .4 0 .6 0 .8 1

C S

0 .5 0 .4 0 .3 0 .2 0 .1 0 0 .1 0 .2 0 .3 0 .4 0 .5

Figure 8: Water surface at 50 hours.

4.1.1. Error estimates and Stability condition

Accuracy and comparison with methods of the results are computed using theL2-norm andL_∞-norm given by:

ErrorL_∞ =max|Φ^Analytic−Φ^RBF|, ErrorL2 =^N

j=1

|(Φ^Analytic−Φ^RBF)²|∆x∆y1/2

, and relative error inL₂-norm,

RE=

N

j=1|(Φ^Analytic−Φ^RBF)²|∆x∆y1/2

N

j=1|Φ^Analytic|∆x∆y ,

whereΦ^Analytic is the analytical solution,Φ^RBF is the simulated one usingRBF smethod. It is well established that for stability,∆thas to be selected at each time step according to the Courant-Friedrichs- Lewy CFL condition

∆t ≤C dmin

max(√

U±gH,√

V ±gH), (4.13)

wheredminis the minimum distance between any two adjacent collocation points,gis the gravitational acceleration andCis the courant number set to be less than unity. Figures 9 - 10 show the comparative error between RBF methods with analytical solution of linear SWE in27hours.

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0 2 4 6 8 10 12 14 16 18 20 0. 1

0 0.1 0.2 0.3 0.4

T ime Hour

Error Tidal (h)

MQ IMQ G S S C S

10⁰ 10¹

10^-8 10^-7 10^-6

T ime Hour MQ IMQ G S S C S

Figure 9: Error comparisonη, (left) usingL_∞-norm, (right) usingL₂relative error in27h.

0 2 4 6 8 10 12 14 16 18 20

0. 1 0.05

0 0.05 0.1 0.15 0.2

T ime Hour

Error Velocity (u)

MQ IMQ G S S C S

10⁰ 10¹

10^-9 10^-8 10^-7 10^-6 10^-5

T ime Hour

MQ IMQ G S S C S

Figure 10: Error comparison foru, (left) usingL_∞-norm, (right) usingL2 relative error in27h.

4.2. Non-linear shallow water equations

We consider the following SWE ([1]):











∂η

∂t +∂(U H)

∂x +∂(V H)

∂y = 0,

∂U

∂t +U∂U

∂x +V ∂U

∂y +g∂η

∂x = 1 Hρω

∂

∂x

νtH∂U

∂x

+ 1 Hρω

∂

∂y

νtH∂U

∂y +g(S0x−Sf x) +f V + τx

Hρω

,

∂V

∂t +U∂V

∂x +V ∂V

∂y +g∂η

∂y = 1 Hρω

∂

∂x

ν_tH∂V

∂x

+ 1 Hρω

∂

∂y

ν_tH∂V

∂y +g(S0y−Sf y)−f U + τy

Hρω

,

(4.14)

wherexand yare the horizontal coordinates, t is the time, H = H(x, y, t)is the instantaneous total depth of water, such thatH = h+η,η = η(x, y, t)is the free surface elevation, his the mean depth of water level, U = U(x, y, t) is the depth-averaged velocity in the x direction, V = V(x, y, t) is the depth-averaged velocity in the y direction, S0x and S0y are the bottom bed slopes deﬁned

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by : S0x = −∂zf

∂x and S0y = −∂zf

∂y , zf being the bed elevation, Sf x and Sf y correspond to the bottom friction slopes, which are approximated by the Manning formula in the following form : Sf x = n²U√

U²+V²

H^4/3 ,Sf y = n²V√

U²+V²

H^4/3 ,wheren is the Manning roughness coefﬁcient,τx and τyare the wind stresses given by:τx =λWx

W_x²+W_y², τy =λWy

W_x²+W_y²,hereλis the surface friction coefﬁcient,Wx andWy are the wind stresses in thexand ydirections, f is the coriolis force parameter, g is the gravitational acceleration, ρ_ω is the density of water,ν_t is the kinematic viscosity.

Some of these parameters are given in the following table :

Parameter n f g ρω ν

Value 0.01 8.55×10⁻⁵ 9.81 1000 10⁻³ Table 1: Constants coefﬁcients.

4.2.1. Real case: Ourika valley ﬂood

For the numerical experiment, the proposed model is applied to the realistic situation of the Ourika valley that is located in the high Atlas mountain some forty kilometers south of the city of Marrakech in Morocco. The flood occurred on August17,1995. We simulated the flood wave in order to compute the water depth profiles, velocities, wave time arrivals at different times of simulation. As no detailed digital elevation model is available, we used transverse profiles of the river given at 15 stations separated by200mon average. The channel considered in this work is 2800m long. The hydrograph of flow’s rates measured at time of the flood at the station of Aghbalou (the upstream of the catchment area) is used as an upstream boundary condition. The other calculation parameters have been assessed on the basis of regional and bibliographical information. Notice that the same study was considered in [3], the proposed mathematical model was discretized using finite volume method on unstructured mesh 2D grid with 1102 elements and 609 nodes. In our study we considered only 68 collocation points and we interpolated the solution on609nodes given by unstructured mesh. The geometry of the channel, collocation points and interpolation points are provided in Figure 11. The hydrograph of flow

0 500 1000 1500 2000 2500 3000

1000 800 600 400 200 0 200 400 600 800 1000

C olloca tion points Interp P oints

Figure 11: Collocation points (blue) and interpolation nodes (red).

measured at the time of the ﬂood in the station of Aghbalou is used as an upstream boundary condition.

The instantaneous measured ﬂows are represented in Figure 12. The following data concerning the geometry of the channel, initial and boundary conditions are taken into account.

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0 10 20 30 40 50 0

200 400 600 800 1000 1200

T ime hrs 0. 50 10 20 30 40 50

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

T ime hrs

F low Q (t)(m3/s ) in 17 a nd 18081995 E leva tion (m)

Mea n V elocities (m/s)

Figure 12: Flow hydrograph, Elevation and Mean velocities on upstream section in 17 and 18-08-1995.

• Initial time step : ∆t = 0.01s. CFL value= 0.1, H = 10manywhere in the channel att = 0. η = 0m, U = −0.29m/s and V = 0m/s at t = 0. At the inlet located at x = 3000m: mean velocities are given in Figure 12 as a boundary condition. At the solid boundaries: normal velocities are taken equal to zero. At the outlet located atx = 200m: the Newmann condition.

∂nη =∂nU =∂nV = 0is considered.

Figure 13 show the evolution of the water depth (free surface in 3D) for some times of the simulation.

We remark clearly the evolution of the flooding wave. However, in Figures 14, the amplitude of the elevation is important around the times where the flood occurred exactly at20h30, fort= 22h, passing the period of the flood the amplitude of the elevation decrease and becomes normal. It should be noted that the results found are practically identical to those found in [3], where the authors used the finite volume method on an unstructured mesh of1102triangles. The important point is that the RBF used in this paper need only some collocation points. So the computational cost is very low compared to the one obtained using the finite volume method.

0

2000 1000

0 1000

10 12 14 16

10 10. 005 10. 01 10. 015 10. 02

0

2000 1000

0 1000

10 12 14 16

10 10.1 10.2 10.3 10.4

0

2000 1000

0 1000

10 12 14 16

10 10.2 10.4 10.6

Figure 13: Depth proﬁle at20,1000and3600second.

Figure 14: Depth proﬁle at20h,20h30min and20h40min.

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5. Conclusion

In this paper, we set-up some efficient algorithms based on Kansa’s RBF and Wendland’s CSRBF for simulating large scale time-dependent problems for SWE. The RBFs are employed for spatial approximation in conjunction with an explicit forward finite difference scheme in time. The proposed scheme is first applied to classical Burger’s equation in 2-D, and to linear SWE in a rectangular channel for validation purposes. A real case study was then investigated. It concerns the Ourika Valley in Morocco threatened by the 1995 flood which injured more than 1300 people and made significant damages to rebuild the inundated areas and understand the flood wave propagation mechanisms. The results obtained with simplicity and very small computation CPU time, for this realistic application show the consis- tency and robustness of the suggested numerical model and that it could be used reliably in Ourika or other real field sites for further events simulations.

Acknowledgements

The authors acknowledge the sponsorship of Hydro3+3 Euromed project and Moroccan CNRST.

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