ABSTRACT
In this chapter, the author describes the differential equation in time and he must approximate the differentiation by finite differences. In addition, when the author chooses a finite difference scheme to describe the differential operator, he considers accuracy and economy in computation. The author demonstrates that the stability of this numerical scheme, they shall take the harmonic oscillator as an example. He discusses the particle pushing in a magnetic field at greater length. The Runge-Kutta method is a popular and powerful method for integrating nonlinear differential equations. One finds that for non-dissipative systems leapfrog trajectories close on themselves whereas Runge-Kutta trajectories gain energy at a rate determined by the allowed error. The transport phenomena in plasma physics is very important and involves the diffusion process. The simplest way of solving the diffusion equation in time is to use a first-order explicit method, which is analogous to the Euler method for ordinary differential equations.
