The Smoothed Particle Hydrodynamics Method For Solving Shallow Water Equations

With the developments of ocean exploitation, disaster preparedness and environmental protection, more and more interest is focus on the numerical solution of the shallow water equations, which has become a hot topic in the international research areas of the Computational Mathematics and Computational Fluid Dynamics.In recent years, people always use the grid-based solutions to solve shallow water equations. In this paper, the smoothed particle hydrodynamics(SPH) which is a meshfree particle method is introduced. The SPH model for solving the shallow water equation is established on the basis of the SPH basic ideas. And the information on the methods and procedures which are involved in using SPH method are given in detail.In this paper, a distortion of Monaghan artificial viscosity is introduced to prevent unphysical penetration for particles approaching each other and remove numerical oscillation in hydrodynamics simulations using SPH. One approach to dynamically evolve the smoothing length is proposed to keep the number of neighboring particles relatively constant and improve the efficiency of the computation and the accuracy of the solution. Meanwhile, in order to preserve the symmetry of particle interaction, some ways to perform the modification are used to produce a symmetric smoothing length.Two types of virtual particles which Liu suggested are introduced to solve the boundary deficiency problem and treat the solid boundary conditions. When a real particle is approaching the boundary, a force is applied pair-wisely along the centering line connecting the real particle and a particle of the first type which is the neighboring particle of the real particle. The particles of the second type are used to treat the boundary deficiency problem, improve the accuracy of the SPH approximation in the boundary region and prevent the unphysical particle penetration outside the solid boundary. Moreover, some existed nearest neighboring particle searching is introduced in detail and discussed preliminarily. The all-pair search algorithm and tree search algorithm are carried out and used in simulating the one-dimensional and two-dimensional dam-break problems. Some useful conclusions are obtained through comparing the time consumed.In this paper, some familiar liquid phenomena are simulated, such as water droplets, the process of the wind blowing water surface. The classical problem-dam-break problem in the computational fluid dynamics field is also simulated, and good results are achieved. In addition, a law of the initial particle distribution is obtained through the experiment, i.e., making the quality of each particle equal as far as possible or change continuously. Furthermore, the results obtained by SPH method are compared with that obtained by the Lagrangian grid-based methods, such as discontinuous Galerkin method which uses the adaptive grid, point out the deficiency which still existed in the simulation by SPH method.