The Linked Cell Method for Short-Range Potentials

Abstract

In Chapter 1 we introduced the particle model, first potential functions and the basic algorithm. Further potentials were presented in Section 2.2.4. So far, we left open how to evaluate the potentials or forces efficiently and how to choose a suitable time integration method. The following chapters will cover these issues. Note that the different methods and algorithms for the evaluation of the forces depend strongly on the kind of the potential used in the model. We will start in this chapter with the derivation of an algorithm for short-range interactions. This approach exploits the fast decay of a short-range potential function and the associated forces. Thus, short-range interactions can be approximated well if only the geometrically closest neighbors of each particle are considered. Note that the algorithm presented here also forms the basis for the methods discussed in the subsequent Chapters 7 and 8 for problems with long-range interactions. We now consider a system which consists of N particles with masses {m 1 , · · · , m N } characterized by the positions {x 1 ,. .. , x N } and the associated velocities {v 1 , · · · , v N } (respective momenta p i = m i v i). x i and v i are here two-dimensional or three-dimensional vectors (one dimension for each direction) and are functions of time t. The space spanned by the degrees of freedom for the positions and velocities is called phase space. Each point in the 4N-dimensional or 6N-dimensional phase space represents a particular configuration of the system. We assume that the domain of the simulation is rectangular, that is, Ω = [0, L 1 ] × [0, L 2 ] in two dimensions, and Ω = [0, L 1 ] × [0, L 2 ] × [0, L 3 ] in three dimensions, respectively, with sides of the lengths L 1 , L 2 , and L 3. Depending on the specific problem certain conditions are imposed on the boundary that are introduced in the following without giving too many details at first. A more substantial description of these boundary conditions can be found in the subsequent application sections. In periodic systems, as for example in crystals, it is natural to impose periodicity conditions on the boundaries. Periodic conditions are also used in non-periodic problems to compensate for the limited size of a numerical simulation domain Ω. In that case, the system is extended artificially by periodic continuation to the entire R 2 or R 3 , respectively, compare Figure 3.1. Particles that leave the domain at one side reenter the domain at the opposite