Numerical solution of bed load transport equations using discrete least squares meshless (DLSM) method. (English) Zbl 1481.76250
Summary: The discrete least squares meshless (DLSM) method is extended in this paper for solving coupled bedload sediment transport equations. The mathematical formulation of this model consists of shallow water equations for the hydrodynamical component and an Exner equation expressing sediment continuity for the bedload transport. This method uses the moving least squares (MLS) function approximation to construct the shape functions and the minimizing least squares functional method to discretize the system of equations. The method can be viewed as a truly meshless method as it does not need any mesh for both field variable approximation and the construction of system matrices; it also provides the symmetric coefficient matrix. In the present work, several benchmark problems are studied and compared with the work of other researchers; the proposed method shows good accuracy, high convergence rate, and high efficiency, even for irregularly distributed nodes. At the end, a real test problem is performed to show and verify the main benefit and applicability of the proposed method to cope with complex geometry in practical problems.
References:
| [1] | Grass, A., Sediment Transport by Waves and Currents (1981), Department of Civil Engineering, University College: Department of Civil Engineering, University College London |
| [2] | Meyer-Peter, E.; Müller, R., Formulas for bed-load transport, (Proceedings of the IAHSR 2nd Meeting (1948), IAHR: IAHR Stockholm), appendix 2 |
| [3] | Van Rijn, L. C., Sediment transport, part I: bed load transport, J. Hydraul. Eng., 110, 10, 1431-1456 (1984) |
| [4] | Einstein, H. A., Formulas for the transportation of bed load, Trans. ASCE Paper, 2140, 561-597 (1942) |
| [5] | Peña González, E., Estudio numérico y experimental del transporte de sedimentos en cauces aluviales. 2002. |
| [6] | Castro Díaz, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M., Sediment transport models in shallow water equations and numerical approach by high order finite volume methods, Comput. Fluids, 37, 3, 299-316 (2008) · Zbl 1237.76082 |
| [7] | Hudson, J.; Sweby, P. K., Formulations for numerically approximating hyperbolic systems governing sediment transport, J. Sci. Comput., 19, 1-3, 225-252 (2003) · Zbl 1081.76572 |
| [8] | Dı, M. C., Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes., Comput. Methods Appl. Mech. Eng., 198, 33-36, 2520-2538 (2009) · Zbl 1228.76091 |
| [9] | Benkhaldoun, F.; Sahmim, S.; Seaid, M., Solution of the sediment transport equations using a finite volume method based on sign matrix, SIAM J. Sci. Comput., 31, 4, 2866-2889 (2009) · Zbl 1201.35021 |
| [10] | Benkhaldoun, F.; Sahmim, S.; Seaid, M., A two‐dimensional finite volume morphodynamic model on unstructured triangular grids, Int. J. Numer. Methods Fluids, 63, 11, 1296-1327 (2010) · Zbl 1425.76155 |
| [11] | Hudson, J., Numerical Techniques for Morphodynamic Modelling (2001), University of Reading |
| [12] | Vetsch, D.F., Numerical simulation of sediment transport with meshfree methods. 2011. |
| [13] | Firoozjaee, A. R.; Sahebdel, M., Element-free Galerkin method for numerical simulation of sediment transport equations on regular and irregular distribution of nodes, Eng. Anal. Bound. Elem., 84, 108-116 (2017) · Zbl 1403.76049 |
| [14] | Liu, G.-. R., Meshfree Methods: Moving Beyond the Finite Element Method (2009), CRC Press |
| [15] | Malidareh, B. F.; Hosseini, S. A., Collocated discrete subdomain meshless method for dam-break and dam-breaching modelling, (Proceedings of the Institution of Civil Engineers Water Management (2017), Thomas Telford Ltd) |
| [16] | Arzani, H.; Afshar, M., Solution of spillways flow by discrete least square meshless methods, (Proceedings of Second ECCOMAS Thematic Conference on Meshless Methods (2007)) |
| [17] | Firoozjaee, A. R.; Afshar, M. H., Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations, Eng. Anal. Bound. Elem., 33, 1, 83-92 (2009) · Zbl 1160.65339 |
| [18] | Firoozjaee, A. R.; Afshar, M., Discrete least squares meshless (DLSM) method for simulation of steady state shallow water flows, Scientia Iranica, 18, 4, 835-845 (2011) · Zbl 1277.76041 |
| [19] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 155, 141-158 (1981) · Zbl 0469.41005 |
| [20] | Izem, N.; Seaid, M.; Wakrim, M., A high-order nodal discontinuous galerkin method for 1D morphodynamic modelling, Int. J. Comput. Appl., 41, 15 (2012) |
| [21] | Benkhaldoun, F.; Seaïd, M.; Sahmim, S., Mathematical development and verification of a finite volume model for morphodynamic flow applications, Adv. Appl. Math. Mech., 3, 4, 470-492 (2011) · Zbl 1262.65110 |
| [22] | Bilanceri, M., Linearized implicit time advancing and defect correction applied to sediment transport simulations, Comput. Fluids, 63, 82-104 (2012) · Zbl 1365.76138 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
