Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> A. NEAREST NEIGHBOR PARTICLE APPROACH <s> Abstract The particle method SPH is applied to one-dimensional shock tube problems by incorporating an artificial viscosity into the equations of motion. When the artificial viscosity is either a bulk viscosity or the Von Neumann-Richtmyer viscosity, in a form analogous to that for finite differences, the results show either excessive oscillation or excessive smearing of the shock front. The reason for the excessive particle oscillation is that, in the standard form, the artificial viscosity cannot dampen irregular motion on the scale of the particle separation since that scale is usually less than the resolution of the interpolating kernel. We propose a new form of artificial viscosity which eliminates this problem. The resulting shock simulation has negligible oscillation and satisfactorily sharp discontinuities. Results with a gaussian interpolating kernel (with second-order errors) are shown to be greatly inferior to those with a super gaussian kernel (with fourth-order errors). <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> A. NEAREST NEIGHBOR PARTICLE APPROACH <s> A new, general-purpose code for evolving three-dimensional, self-gravitating fluids in astrophyics, both with and without collisionless matter, is described. In this TREESPH code, hydrodynamic properties are determined using a Monte Carlo-like approach known as smoothed particle hydrodynamics (SPH). Unlike most previous implementations of SPH, gravitational forces are computed with a hierarchical tree algorithm. Multiple expansions are used to approximate the potential of distant groups of particles, reducing the cost per step. More significantly, the improvement in efficiency is achieved without the introduction of a grid. A unification of SPH with the hierarchical tree method is a natural way of allowing for larger N within a Lagrangian framework. The data structures used to manipulate the grouping of particles can be applied directly to certain aspects of the SPH calculation. 75 refs. <s> BIB002
Nearest neighbor search: in SPH, the calculation of the related examples is only related to the particles in the support domain due to the presence of support domains. The particles that are included in these support domains are called the nearest neighbor particles (NNP) of the relevant particles. The process of finding the nearest neighbor particle is called nearest neighbor particle search (NNPS)  . The most commonly used NNPS has three types: • full-matching search method • linked list search method • tree search method The full-matching search method calculates the distance between all the particles and particle i in the problem domain, searching for all the part icles in particle i support domain. This search process for all particles has the time complexity of O(n 2 ); thus, the calculation time consumption is large and is not suited to the huge number of particles. In the realization of a linked list search method BIB001 -  , the first problem in the domain lies on a layer of a temporary grid. The grid size should match the size of the supported domain. For given particle i, its adjacent particles can only be in the same grid or in close adjacent units. The list search method distributes each particle in the grid cell and concatenates all the particles within each grid with a simple storage rule; its time complexity is O(n)  . Figure 7 is a schematic diagram of a two-dimensional space linked list search method. The problem with the chain search is that when the smooth length varies, the grid space cannot be adapted to each particle. If the list search method is applied, the search efficiency will be low. The tree search method BIB002 recursively divides the largest problem domain into a hexagram until each of all divisions contains only one particle. For any particle i, within the 2κh cube around it, if the cube of the tested particle does not overlap with the cube range of particle i, the search will be stopped. When overlap occurs, the search will continue to the next layer until only one particle is at the current node. If particle i is in the support domain of the particle, then a neighboring particle of particle i will be found. The order of its time complexity is O(N lgN ). The tree search method is highly efficient for solving problems with various smooth lengths and large numbers of particles; therefore, it is very suitable for smooth length concept SPH fluid simulation, with strong robustness  .