Numerical Methodology

Abstract

It will be necessary in particle modeling to solve the system of second order differential equations (1.6) from given initial data. Since, in general, $\vec F$ will be nonlinear, numerical methodology will be essential. Our choice of numerical methods will be guided by the following two observations. First, since ø depends only on r, the system (1.6) is completely conservative (Goldstein (1980)), that is, the system’s energy, linear momentum, and angular momentum are time invariants. Second, for many potentials and forces to be considered, r = 0 will be a singularity in the sense that the potential or the magnitude of the force becomes unbounded as r goes to zero. For such cases the time step must be small for values of r close to zero, which precludes the basic value of high order numerical techniques. For these reasons we will employ only the two methods to be discussed in this chapter, that is, the leap frog method, which is basically a central difference, low order method which is highly efficient and easy to program, and a completely conservative method which conserves exactly the same invariants as (1.6).