Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow

In this work we are interested in numerical simulations for bedload erosion processes. We present a relaxation solver that we apply to moving dunes test cases in one and two dimensions. In particular we retrieve the so-called anti-dune process that is well described in the experiments. In order to be able to run 2D test cases with reasonable CPU time, we also describe and apply a parallelization procedure by using domain decomposition based on the classical MPI library.Comment: 19 page

Metode MacCormack untuk menyelesaikan model transpor sedimen permukaan dasar satu dimensi

In this work, we investigate the numerical solution of one-dimensional bed-load sediment transport model using two steps finite difference method which so-called MacCormack method. Bed-load sediment transport model is composed by the shallow water equation and Exner equation. The Meyer-Peter and Muller (MPM) formula and Wu formula will be used to determine the Grass factor of the bed-load sediment transport. These governing equations will be discretized into predictor and corrector steps of the MacCormack method. The numerical results of the MacCormack method will be validated with an analytical solution of the bed-load sediment transport model. In addition, the MacCormack solution will also be compared with experimental solutions and another numerical method solutions that have existed previously. The numerical results based on MacCormack method give excellent results in which the numerical and the analytical results are hardly differentiated with RMSE of around 00042  or 4,2 .Penelitian ini menyelidiki solusi numerik model transpor sedimen permukaan dasar menggunakan metode beda hingga dua step yang disebut metode MacCormack. Model transpor sedimen permukaan dasar dibangun atas persamaan air dangkal dan persamaan Exner. Formula Meyer-Peter dan Muller (MPM) dan formula wu akan digunakan untuk menentukan faktor Grass dari transpor sedimen permukaan dasar. Persamaan pembangun ini didiskritisasi kedalam step prediktor dan korektor dari metode MacCormack. Hasil numerik metode MacCormack akan divalidasi dengan sebuah solusi analitik dari model transpor sedimen permukaan dasar. Selain itu, solusi metode MacCormack juga akan dibandingkan dengan solusi-solusi eksperimen dan solusi-solusi metode numerik yang telah ada sebelumnya. Hasil numerik berdasarkan metode MacCormack memberikan hasil yang sangat baik dimana hasil numerik dan hasil analitik hampir tidak dapat dibedakan

Modeling Shallow Water Flows on General Terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures

A finite volume shock-capturing solver of the fully coupled shallow water-sediment equations

This paper describes a numerical solver of well-balanced, 2D depth-averaged shallow water-sediment equations. The equations permit variable variable horizontal fluid density and are designed to model watersediment flow over a mobile bed. A Godunov-type, HLLC finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws which describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi-analytical solutions for bedload transport and suspended sediment transport, respectively. The well-balanced property of the equations is verified for a variable-density dam break flow over discontinuous bathymetry. Simulations of an idealised dam-break flow over an erodible bed are in excellent agreement with previously published results ([1]), validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable-density governing equations. Flow hydrodynamics and final bed topography of a laboratory-based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water-sediment models to the choice of closure relationships

A 1D numerical model for the simulation of unsteady and highly erosive flows in rivers

This work is focused on a numerical finite volume scheme for the coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expressions modeling the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled equations can be rewritten as a non-conservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. Moreover, the wave celerities for the coupled morpho-hydrodyamical system depend on the erosion-deposition mechanism selected to update the channel cross-section profile. This influence is incorporated into the system solution by means of a new parameter related to the channel bottom variation celerity. Special interest is put to show that, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge in presence of arbitrary shape channels. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver, first order accurate in time and space, dealing with solid transport flux variations caused by the channel geometry changes. Channel cross-section variations lead to the appearance of a new solid flux source term which should be discretized properly. The stability region is controlled by wave celerities together with a proper reconstruction of the approximate local Riemann problem solution, enforcing positive values for the intermediate states of the conserved variables. Comparison of the numerical results for several analytical and experimental cases demonstrates the effectiveness, exact well-balancedness and accuracy of the scheme

An unstructured finite-volume method for coupled models of suspended sediment and bed load transport in shallow-water flows

The aim of this work is to develop a well-balanced finite-volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two-dimensional shallow-water flows. The modelling system consists of three coupled model components: (i) the shallow-water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite-volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite-volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam-break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon