Study On The Shallow Water Equations Of High Resolution Algorithm

Based on the finite volume method and Roe’s approximate Riemann solver, the One-dimensional and Two-dimensional conservative high-resolution shallow water models with source terms are developed in this study.The discretization of the bed slope source terms is done following an upwind approach to protect the scheme harmonious. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model ensures a global conservation and positive values of both water level and solute concentration. The main work of this article is as following:1. A technique has been investigated for extending Roe’s finite volume method to second-order spatial and temporal accurate MUSCL-type slope-limiting approach in order to simulate shallow water flow over uneven beds. The discretization of the bed slope source terms is done following an upwind approach to ensure convective term from left side of the equation and bottom slope source term from right side of the equation to preserve the "humanity" of conservative numerical scheme. The objective of this study is to compare the performances corresponding to different variables of reconstruction to determine whether there exists all optimal approach.2. A Delaunay mesh generation method based on Bowyer-Watson method is designed for a complex region. Based on the initial mesh, a edge recovery algorithm by no adding Steiner points is presented. A new technique considering the order and the location of insertion points is established to generate high quality mesh. Many methods are adopted to improve the quality and efficiency of the mesh.3. A wetting/drying condition (WDC) for unsteady shallow water flow leading to zero mass error is presented. The WDC has been incorporated into a cell-centred finite volume method based on Roe’s approximate Riemann solver on unstructured grids. The discretization of the bed slope source terms was done following all upwind approach and the semi-implicit treatment was used for the friction source terms. It is shown that the numerical technique of improving bed slope and limited flux style can exactly reproduce steady state of still water and enable the model to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. A new PLCD method based on two-dimensional MUSCL-type finite volume schemes is developed. Numerical results are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real complex geometries and bottom slope variation.4. A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. This model corrects solute concentration to keep the results reasonable. This paper discusses the difference of diffusion terms between explicit and implicit discretization. The centered discretization of the diffusion terms is in an implicit way in this model. Numerical experiments show that this model has good stability and smoothness and ensures conservation of the quality of water and solute concentration. Therefore, it could be used for the simulation of solute transport.5. A PLCD (Project Limited Central Difference) technique has been investigated based on MUSCL-type slope-limiting approach. The performances corresponding to different variables of reconstruction (LCD, Durlofsky, PLCD, MLG and MLG-Wierse) are compared. Numerical results indicate that PLCD, MLG and MLG-Wierse are efficient and the new PLCD computation expense of the new constructed PLCD scheme is much less than MLG and MLG-Wierse.