It is usually more difficult to simulate continuous systems than discrete ones, especially when the properties under study are governed by nonlinear equations. The systems can be so complex that the length scale at the atomic level can be as important as the length scale at the macroscopic level. The basic idea in dealing with complicated systems is similar to a divide-and-conquer concept, that is, dividing the systems with an understanding of the length scales involved and then solving the problem with an appropriate method at each length scale. A specific length scale is usually associated with an energy scale, such as the average temperature of the system or the average interaction of each pair of particles. The divide-and-conquer schemes are quite powerful in a wide range of applications. However, each method has its advantages and disadvantages, depending on the particular system.

Hydrodynamic equations

In this chapter, we will discuss several methods used in simulating continuous systems. First we will discuss a quite mature method, the finite element method, which sets up the idea of partitioning the system according to physical condition. Then we will discuss another method, the particle-in-cell method, which adopts a mean-field concept in dealing with a large system involving many, many atoms, for example, 1023 atoms. This method has been very successful in the simulations of plasma, galactic, hydrodynamic, and magnetohydrodynamic systems.