The evaluation of sediment transport and the involved processes is a major issue inhydraulic and river engineering. Soil erosion by water is a complex phenomenon that isaffected by many factors such as climate, topography, soil characteristics, vegetation, andanthropogenic activities. The ability to design numerical methods able to predict themorphodynamic evolution of the bed level has clear mathematical and engineering utilities.In most sediment-transport problems, the mathematical model is a coupled model that iscomposed by a hydrodynamical and a morphodynamical component, which relies on asystem of four equations for two-dimensional problems and three equations for one-dimensional problems: two or three equations (shallow water equations) describes flowcontinuity and momentum conservation with shallow water assumptions (hydrodynamicalcomponent), and one equation (the Exner equation) expresses sediment continuity(morphodynamical component). In literature, different equations to model the solidtransport sediment flux can be found. The Grass equation 1, Meyer-Peter & Muller’sequation 2, Van Rijn’s equation 3, Einstein’s equation 4, etc., can generally be obtainedby empirical methods.They are system of nonlinear partial differential equations (PDEs); thus, they generally haveno analytical solutions and, therefore, we have to rely on numerical solutions. It is of greatimportance to obtain an accurate numerical approximation of the governing equations.Various numerical methods developed for the general systems of hyperbolic conservationlaws have been applied to morphodynamic systems. The finite-volume method has beenpaid significant attention in recent decades.Pena Gonzalez proposed a numerical approximation of a non-coupled system formed byshallow water equations and the bedload sediment transport equation. He applied a finitevolume method based on the Roe scheme to obtain the shallow water unknowns, i.e., thethickness and discharge of the fluid. The approximated velocity and the thickness of thefluid are used in the bedload sediment transport equation to get an approximation of the3thickness of the sediment layer. He also proposed different techniques of measurement tocalculate the thickness of the sediment layer 5.Coupling between the shallow water and the Exner equations introduces additionalmathematical and numerical difficulties, which have also been rather extensively studied inthe literature (e.g., the studies and the references in 6-10).Hudson proposed several numerical approximations of a system constituted by the shallowwater equations and bedload sediment transport equation using the Grass model. Heconsiders different reformulations of the problem with steady and unsteady approaches;this means that he has considered both the coupled and the non-coupled system ofsediment-transport equations. A variety of the numerical schemes discussed included theadapted versions of the Lax–Friedrichs scheme, the classic Lax–Wendroff scheme, theMacCormack scheme, and Roe’s scheme. The high-resolution schemes that were derivedalso satisfy the TVD property so that no spurious oscillations occur in the numerical resultsin 11.In the last three decades, the meshless methods have been significantly developed. Anumerical model for simulation, which is based on two meshless particle methods, ispresented in 12. The fluid is modeled by a continuum approach, which is discretized by thesmoothed particle hydrodynamics (SPH) method. The sediment articles are represented bythe discrete element method (DEM), where the interactions between the discrete sedimentgrains are modeled by a force law, which is also able to account for the various kinds offriction. The numerical software used in this work is called PASIMODO (“particle simulationand molecular dynamics in an object oriented fashion”); this is a multipurpose particle-simulation tool. It is, however, not comparable to our method because the method that isused in this article follows the Lagrangian point of view and is particle-based.Recently, the element-free Galerkin (EFG) meshless method was developed for thesimulation of sediment transport in 2 dimensions 13. They formulated these equations in acoupled approach and rewrote them as non-conservative hyperbolic systems.Among all the previous works, all of them need pre-defined meshes to discretized spatialdomain. Even the EFG, however, is not a truly meshless method since the integrals areevaluated on background cell structures and the coefficient matrix will be asymmetric.Fazli Malidareh and Hosseini introduced a collocated discrete subdomain meshless methodto solve shallow-water equations for free-surface flows with the conservation of sediment4mass. They have, however, only provided this method for solving the 1D dam break and thedam breach model, with regular node distribution and provide the asymmetric coefficientmatrix 14.In the present work, a numerical model based on meshless methods is applied, which isstable, reliable, and accurate. This meshless method, namely the discrete least squaresmeshless (DLSM) method was proposed by Arzani and Afshar 15, and used for the solutionof shallow-water equations on regularly distributed nodes. The method, however, sufferedfrom a lack of accuracy on the irregular distributions of nodes. The method was latermodified by Firoozjaee and Afshar, who introduced a set of new points, named samplingpoints 16.The method used a full least squares approach in both the function approximation and thediscretization of governing differential equations in space. The meshless shape functions arederived using the moving least squares (MLS) method of function approximation. Thediscretized equations are obtained via a discrete least squares method, in which theequations are based on minimizing the least squares functional, which is defined by the sumof squared residuals of the differential equations and its boundary condition at some points(called sampling points) over the domain and its boundary. The sampling points can begenerally considered to be different from the nodal points used to discretize the problemdomain. This method can be viewed as a truly meshless method as it does not need anymesh for both field variable approximation and the construction of system matrices; it alsoprovides the symmetric coefficient matrix, and preserves high accuracy and high stabilityeven for irregularly distributed nodes.In 16 the efficiency and the accuracy of the DLSM, the method is tested against some one-and two-dimensional elliptic problems on regular and irregular meshes of nodes and theresults are presented and compared with the exact solutions.Firoozjaee and Afshar, in 17, used the DLSM method for the solution of shallow-waterproblems. Certain 2D benchmark problems and a 2D flow over an ogee spillway were usedto illustrate the performance of the DLSM method. The results of the numerical examples ofshallow-water problems show good quality for irregular node positioning.In this article, the DLSM method is extended to solve sediment transport problems. Thediscretization of equilibrium equations in the sediment transport, using the DLSM method,is described in Section 2. Section 3 states a brief description of the DLSM method. Numerical5examples are demonstrated in Section 4. Certain concluding remarks are presented inSection 5.

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