1-1 INTRODUCTION
In this book, we shall be concerned with an area of scientific investigation where the computer plays a central role. The topics we shall consider are usually classified under the heading “computational physics,” although such a title is a little misleading since the material covered overlaps many established subject areas: physics, chemistry, engineering, numerical analysis, and computer science all play a part in computational physics. For this reason, we prefer the more general term “computational science.”
The starting point of the computational science approach to scientific investigation is a mathematical model of the physical phenomenon of interest. The equations of the mathematical model are cast into a discrete algebraic form which is amenable to numerical solution. The discrete algebraic equations describe the simulation model which, when expressed as a sequence of computer instructions, provides the computer simulation program. The computer plus program then allow the evolution of the model physical system to be investigated in computer experiments.
Computer simulation may be regarded as the theoretical exercise of numerically solving an initial-value–boundary-value problem. At time t = 0, the initial state of the system is specified in some finite region of space (the computational box) on the surface of which prescribed boundary conditions hold. The simulation consists of following the temporal evolution of the configuration. The main part of the calculation is the timestep cycle in which the state of the physical system is incremented forwards in time by a small timestep, DT. The experimental aspect, and thence the name computer experiment, arises when we consider the problems of measurements. Even the simplest simulation calculation generates large amounts of data which require an experimental approach to obtain results in a digestible form.
Although the amount of data which can be handled by computers is large, it is nevertheless finite. Much of the ingenuity of computational scientists is devoted to obtaining good simulation models of the physical systems within the constraints of the available finite computer resources. Methods of discretization used in obtaining simulation models include finite-difference methods (Richtmyer and Morton, 1967), finite-element methods (Strang and Fix, 1973), and the methods to be dealt with in this book, particle methods.
“Particle models” is a generic term for the class of simulation models where the discrete representation of physical phenomena involves the use of interacting particles. The name “particle” arose because in most applications the particles may be identified directly with physical objects. Each particle has a set of attributes, such as mass, charge, vorticity, position, momentum. The state of the physical system is defined by the attributes of a finite ensemble of particles and the evolution of the system is determined by the laws of interactions of the particles. A feature which makes particle models computationally attractive is that a number of the particle attributes are conserved quantities and so need no updating as the computer simulation evolves in time.
The relationship between the particles of the simulation model and particles in physical systems is determined largely by the interplay of finite computer resources and the length and timescale...
