Climate change-induced natural disasters and increased rainfall intensities have become frequent over the past few decades [1] [2] . Among these disasters, floods and inundations have emerged as highly destructive in urban areas, causing significant human and economic losses [3] [4] . According to the United Nations International Strategy for Disaster Reduction report [5] , between 1994 and 2013, the Emergency Events Database recorded 6873 natural disasters worldwide, resulting in the loss of 1.35 million lives, averaging nearly 68,000 lives per year. Additionally, an average of 218 million people were affected by natural disasters annually during this period. The last report [6] reports that between 2000 and 2019, the number of major floods has more than doubled, rising from 1389 to 3254, while the incidence of storms increased from 1457 to 2034. Floods and storms were the most prevalent events during this period. This phenomenon has sparked worldwide public, political, and scientific interest, highlighting the need for improved flood risk awareness and prevention strategies [7] - [9] . Despite advances in forecasting, management, and defense technologies, floods remain a substantial global threat [10] .

During the recent twenty years, the Comoros Islands have borne the brunt of impactful climate change and variable weather patterns. Notably, heavy rains adversely affected over 70% of the Mohelian population in April 2004 and April 2012, damaging primarily concentrated areas in the island’s northern regions due to inundations and vegetation destruction [11] . These extreme climatic events also consistently devastate livestock, demolish homes, and decimate significant hectares of crops. In April 2012, the Comoros Interior Ministry declared a “National Disaster” in response to the flash floods and requested international assistance [12] . In April 2019, Tropical Cyclone Kenneth crossed the Comoros archipelago, primarily affecting Grand Comoros and causing subsequent flooding in the islands of Anjouan, Grande-Comore, and Moheli. The heavy rainfall associated with TC Kenneth led to widespread flooding. According to joint rapid assessments conducted by the government and partner agencies, the cyclone resulted in seven deaths, 200 injuries, and 20,000 displaced individuals. Additionally, 3818 houses were destroyed, 7013 houses were damaged, 96 water tanks were destroyed, 465 classrooms were damaged, and 213 of them were destroyed. Six health centers were damaged, and one hospital was flooded. Nearly 80% of crops were destroyed, leading to food shortages and inflation in the prices of staple foods [13] . Recently, between April and June 2024, heavy rainfall has been affecting Comoros, triggering landslides and causing floods that resulted in casualties and damage. According to media reports, a landslide occurred in Mutsamudu City on Anjouan Island, resulting in at least three deaths and two injuries. Additionally, on Grande-Comore Island in north-western Comoros, several houses have been affected by floods and landslides, and the village of Misoudje has been isolated. In the northwest of the small island of Moheli, the village of Hoani was severely impacted: houses were demolished, hectares of fields were damaged, a dike was destroyed, and a football field became part of the riverbed.

Numerical simulations are essential in engineering decision-making to understand flood situations and prevent catastrophes. A great number of works are conducted on understanding, modeling, simulating, describing, and assessing flood inundation situations. For more details, one can read [14] [15] , among others. For example, an explicit description of land surface macrostructures of flood inundation is studied in [16] , while an efficient flood inundation modeling based on a data-driven approach was conducted in [17] . Numerical studies are also conducted using shallow water equations [18] [19] . Mesh-based methods have been widely proposed during the recent decades for urban flood modeling and dam-break flows. A finite volume method in a structured grid has been proposed in [20] to model urban flood situations in Ouagadougou, Burkina Faso, while in [21] , a finite difference MacCormack method is adapted for numerical modeling of dam-break induced flow through multiple buildings in an idealized city. A discontinuous finite element Galerkin method was proposed for dam-break flood over natural rivers [22] . However, the accuracy of these methods is impacted by the quality of the meshes, stabilization techniques, and the solutions to Riemann problems. This limitation hinders their application to solving real-world problems with irregular domains and complex Riemann problems.

Meshfree Radial Basis Functions (RBFs) have gained significant attention in recent years as a powerful tool for solving complex problems in various fields, including fluid dynamics [23] - [25] . One such application addresses dam-break problems, which involve the sudden release of water due to the failure of a dam, leading to complex flow patterns that need to be accurately modeled for effective disaster management and infrastructure design. The advantages of using Radial Basis Functions in solving shallow water problems are manifold, making them an attractive choice for researchers and engineers [26] - [28] . RBFs offer excellent flexibility and adaptability in handling complex geometries and boundary conditions, which are common in dam-break scenarios. Moreover, the mesh-free nature of RBFs eliminates the need for mesh generation and refinement, a process that is often time-consuming and prone to errors in conventional numerical methods. This advantage significantly reduces the computational overhead and simplifies the implementation process, allowing for quicker and more efficient simulations. The lake of using mesh-free radial basis functions is how to choose a suitable shape parameter that leads the scheme to be high order accurate and stable [29] - [31] . The current paper proposes the use of a variable shape parameter selection based on particle swarm optimization tools [32] .

This paper will be presented in the following order: the mathematical formulation of the shallow water equations is presented in their conservative form in Section 2. In contrast, the mesh-free radial basis method is presented and explained in Section 3. Numerical validations illustrating the accuracy and the stability of the proposed methods are presented 4. Then a set of numerical simulations is conducted in Section 5. Finally, a summary and perspectives are drawn in Section 6.

The 2D shallow water equations are utilized to model flood inundation. These equations are derived from the conservation of mass and momentum. They form a set of three equations that describe the behavior of the water depth and the horizontal velocities over time. These equations have been widely used to simulate urban flood modeling [3] [4] [20] . The continuity equation, obtained from the equation of the mass conservation, is given by

$\frac{\partial h}{\partial t}+\frac{\partial \left(hu\right)}{\partial x}+\frac{\partial \left(hv\right)}{\partial y}=\mathrm{0,}$(1)

where $h\left[\text{m}\right]$ is the water elevation, $u\left[\text{m}\cdot {\text{s}}^{-1}\right]$ is horizontal velocity in the x component and $v\left[\text{m}\cdot {\text{s}}^{-1}\right]$ is the horizontal velocity according to y coordinate. The momentum conservation equation gives two equations of velocities written by

$\frac{\partial \left(hu\right)}{\partial t}+\frac{\partial }{\partial x}\left(h{u}^2+\frac{1}{2}g{h}^2\right)+\frac{\partial \left(huv\right)}{\partial y}=-gh\frac{\partial z}{\partial x}+S_{f_x}+{\nu }_t\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)\mathrm{,}$(2)

$\frac{\partial \left(hv\right)}{\partial t}+\frac{\partial \left(huv\right)}{\partial x}+\frac{\partial }{\partial y}\left(h{v}^2+\frac{1}{2}g{h}^2\right)=-gh\frac{\partial z}{\partial y}+S_{f_y}+{\nu }_t\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)\mathrm{,}$(3)

where $g\left[\text{m}\cdot {\text{s}}^{-2}\right]$ resents the gravitational acceleration, $z\left[\text{m}\right]$ define the inner bed topography and ${\nu }_t$ is the horizontal viscosity coefficient. The friction terms $S_{f_x}$ and $S_{f_y}$ are computed using Manning’s formula [20] [33] :

$S_{f_x}=-g{n}_m^2\frac{u\sqrt{u^2+v^2}}{h^{4/3}}\text{and}{S}_{f_y}=-g{n}_m^2\frac{v\sqrt{u^2+v^2}}{h^{4/3}}\mathrm{,}$(4)

where $n_m\left[{\text{m}}^{1/3}\cdot {\text{s}}^{-1}\right]$ is the Manning-Strickler roughness coefficient, which characterizes the roughness at the bed. Different values this coefficient can be determined by a GIS-based study and it depends on the characteristics of the material (buildings, trees, soil, …). For more details, one can read for instance [34] and references therein.

Equations (1)-(3) can be written in a compact form as follows:

$\frac{\partial U}{\partial t}+\frac{\partial F\left(U\right)}{\partial x}+\frac{\partial G\left(U\right)}{\partial y}=Q\left(U\right)\mathrm{,}$(5)

where $U$ represents the vector of conservative unknowns, $F$ and $G$ represent the advective and pressure fluxes in the x and y directions, and are given by:

$U=\left(\begin{array}{c}h\\ hu\\ hv\end{array}\right)\mathrm{,}F\left(U\right)=\left(\begin{array}{c}hu\\ h{u}^2+\frac{1}{2}g{h}^2\\ huv\end{array}\right)\mathrm{,}G\left(U\right)=\left(\begin{array}{c}hv\\ huv\\ h{v}^2+\frac{1}{2}g{h}^2\end{array}\right)\mathrm{,}$(6)

and $Q\left(U\right)$, containing the diffusion terms and the friction terms, is expressed by:

$Q\left(U\right)=\left(\begin{array}{c}0\\ -gh\frac{\partial z}{\partial x}+S_{f_x}+{\nu }_t\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)\\ -gh\frac{\partial z}{\partial y}+S_{f_y}+{\nu }_t\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)\end{array}\right)\mathrm{.}$(7)

The numerical modeling of hyperbolic equations such as (1)-(3) requires the design of accurate and efficient tools to capture the fine solution features. Solving numerically these nonlinear equations in their conservative form is still a considerable task in nonlinear hyperbolic situations [35] [36] . Let us consider the compact formulation of the two-dimensional shallow water Equations (5) to be solved in set of $N=N_I+N_B$ collocation points $x_i={\left(x_i\mathrm{,}{y}_i\right)}^{\text{T}}$ where $N_I$ is the number of interior points while $N_B$ is the number of boundary points. For each collocation point $x_i$, the approximation of a solution $U_i\left(t\right)$ is defined as

$U_i\left(t\right)\simeq {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j\left(t\right)\phi \left(r_{ij}\mathrm{,}\epsilon \right)\mathrm{,}$(8)

where $r_{ij}=\Vert x_i-x_j\Vert$ is the Euclidean distance between the points $x_i={\left(x_i\mathrm{,}{y}_i\right)}^{\text{T}}$ and $x_j={\left(x_j\mathrm{,}{y}_j\right)}^{\text{T}}$, $\epsilon \ne 0$ is a shape parameter that controls the fitting of a smooth surface to the data, and $\phi$ is a radial basis function. In the current study, we consider the infinitely smooth multiquadrics radial basis function defined as

$\phi \left(r\mathrm{,}\epsilon \right)=\sqrt{1+{\left(\epsilon r\right)}^2}\mathrm{.}$(9)

The coefficients expansion ${\lambda }_i\left(t\right)$ in (8) are calculated by solving the following linear system of $N\times N$ algebraic equations

$B\Lambda =u\mathrm{,}$(10)

where $B$ is an $N\times N$ matrix with entries $\phi \left(r_{ij}\mathrm{,}\epsilon \right)$, $\Lambda$ and $u$ are N-valued vectors with entries ${\lambda }_i$ and $u_i$, respectively.

The first partial derivatives of (8) at point $x_i$, according to x are obtained by the following equation

$\frac{\partial U_i\left(t\right)}{\partial x}\simeq {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j\left(t\right)\frac{\partial }{\partial x}\phi \left(r_{ij}\mathrm{,}\epsilon \right)\mathrm{.}$(11)

In the same manner, all the spatial derivatives can be expressed in a compact way as:

$\frac{\partial U}{\partial x}=H_x\Lambda ,\frac{\partial U}{\partial y}=H_y\Lambda ,\frac{{\partial }^{2}U}{\partial x^2}=H_{xx}\Lambda ,\frac{{\partial }^{2}U}{\partial y^2}=H_{yy}\Lambda ,$(12)

where $H_x$, $H_y$, $H_{xx}$ and $H_{yy}$ are $N\times N$ matrices with entries $\frac{\partial \phi }{\partial x}\left(r_{ij}\mathrm{,}\epsilon \right)$, $\frac{\partial \phi }{\partial y}\left(r_{ij}\mathrm{,}\epsilon \right)$, $\frac{{\partial }^2\phi }{\partial x^2}\left(r_{ij}\mathrm{,}\epsilon \right)$ and $\frac{{\partial }^2\phi }{\partial y^2}\left(r_{ij}\mathrm{,}\epsilon \right)$, respectively. The above partial derivatives can also be evaluated using a single matrix by vector multiplication as

$\frac{\partial U}{\partial x}=D_{x}U,\frac{\partial U}{\partial y}=D_{y}U,\frac{{\partial }^{2}U}{\partial x^2}=D_{xx}U,\frac{{\partial }^{2}u}{\partial y^2}=D_{yy}U,$(13)

where the derivative matrices $D_x$, $D_{xx}$, $D_y$ and $D_{yy}$ are $N\times N$ matrices given by

$D_x=H_x{B}^{-1},D_y=H_y{B}^{-1},D_{xx}=H_{xx}{B}^{-1},D_{yy}=H_{yy}{B}^{-1}.$(14)

Hence, the RBF approximation of the problem (5) yields the following semi-discrete system

$\frac{\partial U}{\partial t}+D_{x}F\left(U\right)+D_{y}G\left(U\right)=Q\left(U\right)\mathrm{,}$(15)

where $Q\left(U\right)$ is discretized as

$Q\left(U\right)=\left(\begin{array}{c}0\\ -gh{D}_{x}Z+S_{f_x}+{\nu }_t\left(D_{xx}+D_{yy}\right)u\\ -gh{D}_{y}Z+S_{f_y}+{\nu }_t\left(D_{xx}+D_{yy}\right)v\end{array}\right)\mathrm{.}$(16)

The semi-discrete system (15) can also be reformulated in a compact system of ordinary differential equations with the right-hand function

$\frac{\partial U}{\partial t}=ℱ\left(U\right)\mathrm{,}$(17)

where $ℱ\left(U\right)=-D_{x}F\left(U\right)-D_{y}G\left(U\right)+Q\left(U\right)$. Forward Euler method has been mainly used as a time-stepping for RBF methods. However, this method is only first-order accurate in time and it may introduce excessive numerical dissipation in the computed RBF solutions. In the current study, we use an explicit Runge-Kutta method studied in [37] . Hence, to integrate the Equation (17), the time interval is divided into subintervals $\left[t_n\mathrm{,}{t}_{n+1}\right]$ with duration $\Delta t=t_{n+1}-t_n$ for $n=\mathrm{0,1,}\cdots$. The notation $U^n$ denotes the value of a generic function $U$ at time $t_n$. The procedure to advance the solution from the time $t_n$ to the next time $t_{n+1}$ can be carried out as

$\begin{array}{l}{\mathcal{U}}^{\left(1\right)}=U^n+\Delta tF\left(U^n\right)\mathrm{,}\\ {\mathcal{U}}^{\left(2\right)}=\frac{3}{4}{U}^n+\frac{1}{4}{\mathcal{U}}^{\left(1\right)}+\frac{1}{4}\Delta tF\left({\mathcal{U}}^{\left(1\right)}\right)\mathrm{,}\\ U^{n+1}=\frac{1}{3}{U}^n+\frac{2}{3}{\mathcal{U}}^{\left(2\right)}+\frac{2}{3}\Delta tF\left({\mathcal{U}}^{\left(2\right)}\right)\mathrm{.}\end{array}$(18)

The main feature of this method lies on the fact that the progression (18) is a convex combination of first-order Euler steps which exhibits strong stability properties. Therefore, the scheme (18) is TVD, third-order accurate in time, and stable under the usual Courant-Friedrichs-Lewy (CFL) condition.

For all problems involving matrices, solutions depend on the invertibility of the matrix. RBF-based meshless methods mainly rely on the invertibility of the RBF derivative matrix $B$. It has been proven in the literature [23] that invertibility is ensured when using a constant shape parameter. However, this invertibility is lost if the matrix $B$ becomes singular [38] . The shape parameter significantly impacts both the accuracy and stability of the method. When the domain is refined, global shape parameters can lead to ill-conditioned matrices, resulting in compromised accuracy and stability. Recently, techniques have been proposed to enhance the accuracy of the RBF method by using variable shape parameters. A well-known shape parameter in mathematical literature is the exponential shape parameter defined by [29] :

${\epsilon }_i={\left[{\epsilon }_{\min }^2{\left(\frac{{\epsilon }_{\max }^2}{{\epsilon }_{\min }^2}\right)}^{\frac{i-1}{N-1}}\right]}^{1/2},i=1,\cdots ,N.$(19)

The random shape parameter defined by [30] is:

${\epsilon }_i={\epsilon }_{\min }+\left({\epsilon }_{\max }-{\epsilon }_{\min }\right)\times \text{rand}\left(\right)\mathrm{,}$(20)

where rand is the MATLAB function that returns uniformly distributed pseudo-random numbers on the unit interval. The trigonometric shape parameter given by [31] is:

${\epsilon }_i={\epsilon }_{\min }+\left({\epsilon }_{\max }-{\epsilon }_{\min }\right)\times \sin \left(i\right)\mathrm{,}$(21)

where ${\epsilon }_{\min }=\frac{\sqrt{N}}{3}$ and ${\epsilon }_{\max }=\sqrt{N}$, with N being the total number of collocation points. Genetic algorithms are used in [39] to find optimal variable shape parameters for one-dimensional boundary value problems. In the current study, we adopt variable shape parameters for Kansa’s multiquadric using a heuristic optimization method based on particle swarm strategies [32] .

In all the numerical examples considered in this section, the gravitational intensity is fixed to be $g=9.81$; the bottom is assumed to be frictionless ( $n_m=0$), the Courant-Friedrichs-Lewy (CFL) number is taken with respect of the CFL condition $C=0.45$ while time step $\Delta t$ is varied according to the canonical stability condition [40] :

$\Delta t=C\frac{d_{\min }}{\max \left(\sqrt{u\pm gh},\sqrt{v\pm gh}\right)},$(22)

where $d_{\min }$ is the minimum distance between any two adjacent points in the collocation set. After a set of numerical simulations using different value of the viscosity coefficient, this one is taken to be ${\nu }_t=0.0001$ which is suitable to stabilize the considered following examples without introducing notable numerical dissipation.

The case study under consideration was previously examined in [44] for the experimental measurement of dam-break flow against a single isolated obstacle. This case was chosen due to its simplicity, involving only a single building, which facilitates the analysis of the flow behavior around the structure. The experiment was conducted in a flume measuring 35.80 m in length and 3.6 m in width, with a single building positioned in the downstream reservoir. From this study, the Manning coefficient was determined to be $n_m=0.010\text{s}\cdot {\text{m}}^{-1/3}$. The initial conditions of the experiment included a water depth of 0.40 m in the upstream reservoir and 0.02 m in the downstream reservoir. Water depth was monitored at six control points, labeled G1 to G6. This model is solved in [21] using a predictor-corrector MacCormack method coupled with the Hansen numerical Filter. The authors used a comparison of three building approaches to define the buildings and the dam wall (Manning approach, contour approach and wall Boundary approach). Based on the calculation of the Median Absolute Percentage Error (MDAPE) value, their study tried to determine the approach with the highest accuracy. And it has been proved that the wall boundary approach had the highest accuracy. This approach is selected in the current study and only results from [21] using this approach will be used for comparison. To define the building with the wall boundary approach, the velocity perpendicular to the buildings is restricted to zero. The model setup is depicted in Figure 5 . For more details on the position of the Gauges, the Gape, and the building, readers can refer to [44] .

To perform a comparison between the RBF scheme and the MacCormack method, we prefer to use the same uniform grid as in [21] where $\Delta x=0.1$ and $\Delta y=0.1$. The maximum simulation time is set to 30 s with a fixed time step $\Delta t=0.001$. Numerical results are given in Figure 6 where height elevation is shown at different times of simulations. Experimental results and those from [21] are also depicted for view comparison. From a bird’s-eye view, we remark that both RBF and MacCormack methods give similar results for the considered problem. Both capture attentive chocks at the upstream and the downstream of the gate and around the building.

For more comparison, two-dimensional profiles of the instantaneous water elevation at the six gauges are plotted in Figure 7 . The experiment, the RBF, and the MacCormack profiles are provided. Once more, obtained results underline the powerful of the RBF scheme to detect correctly dam-breaks and building inundations, as does the stabilized MacCormack method.

This study presents a novel application of Radial Basis Functions in simulating dam-break flows and urban flood inundation. The RBF-based method offers significant advantages in handling complex geometries and provides accurate and efficient simulations. Furthermore, the results of the water inundation case demonstrate that the RBF method is highly reliable in modeling a more complex 2D scenario, yielding results that closely match the experimental data. The results demonstrate its potential as a valuable tool for urban flood risk management, aiding in the development of effective emergency response strategies.

The authors wish to thank the anonymous referees for their valuable comments that lead to improving the quality of the present work.