###### Abstract  
This paper focuses on the implementation of an accurate meshless scheme for the simulation of the advection-diffusion of non-active pollutant in a two-dimensional depth-averaged flow. The depth-averaged flow model includes the shallow water system and the pollutant propagation described by the advection-diffusion equation with diffusion tensor. The mathematical model was implemented by using a meshless method based on local radial basis functions. This method was used to numerically evaluate the spatial derivatives and complemented with the second-order Runge-Kutta method for the time evolution. To remove the non-physical oscillations, which appear at the discontinuity, a filter based on a hyperviscosity operator was applied. To investigate the effectiveness and accuracy of the proposed scheme, a number of tests are presented, including the dam break problem, and the pure transport of a pollutant in a long channel. Finally, a hypothetical example of a pollutant transport in the Strait of Gibraltar is modeled. The results obtained are compared both with analytical solutions and with simulation results obtained by a finite volume method based on the Roe-MUSCL scheme. The main advantages of the proposed method are: (i) the simplicity of implementation, (ii) the ability to handle calculations of slowly varying flows or concentrations, as well as rapidly varying flows containing shocks or discontinuities, and (iii) the ability to satisfy the C-property and guarantee positive values of both water level and pollutant concentration. shallow water flows; pollutant transport; meshless method; hyperviscosity stability; irregular bed +
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numerical model of shallow water flows, one encounters major difficulties that result as much from the physical complexity of the environment as from the numerical complications. Thus, the geomorphological complexity of rivers, estuaries, and coastal areas often requires narrow computational meshes. Numerical stability then requires a small computational time step, proportional to the size of the space step. The strongly convective flow, possibly with a shock front, causes numerical oscillations or artificial diffusion. These are due to the treatment of the advection terms in the governing equations of the water mass transport, by standard approximation methods. Several methods have been used for solving shallow water equations coupled with the transport-diffusion equation. Most of these were mesh-based methods such as finite differences, finite volumes or finite elements [4; 5; 6]. However, these methods have proven problematic when dealing with high flow or concentration gradients and tend to produce numerical oscillations or numerical diffusion. To deal with these challenges, there are also semi-analytical or mesoscopic approximation methods for the solution of time-dependent partial differential equations [7; 8; 9]. Historically, the Roe scheme was among the first schemes to be adapted to conservation law problems [10; 11], including shallow water flow problems [12]. The Roe scheme has been used for solving the Riemann problem, which naturally arises when applying mesh-based methods. Over the years, several authors have developed new schemes that use the same concepts. These schemes differ either in the way they incorporate the other source terms or in the technique used to increase the order of precision. Among the methods used were the Essentially Non-Oscillatory (ENO) [13] or Weighted Essentially Non-Oscillatory (WENO) [14] methods, which reconstruct the solution using polynomials of arbitrary degree. Some authors have also developed centrally weighted essential non oscillatory (CWENO) schemes [15]. Another approach is the Multi-dimensional Optimal Order Detection method, originally developed in [16]. This method also uses polynomial reconstruction of the solution, with polynomials of as high a degree as desired. Since its introduction, it has been extended by its authors in [17; 18].