Finite Difference Models of the Simple Harmonic Oscillator

The simple harmonic oscillator (SHO) is the basis of many physical models. Since it can be solved analytically, it is a useful vehicle to introduce the basic concepts of the FDTD methodology without being distracted by the technical details needed to implement FDTD for the wave equation and for Maxwell’s equations. In this chapter we derive exact FDTD algorithms from nonstandard FD models to solve the free, damped, and damped and driven SHO. The concepts developed in this chapter will be subsequently extended to develop nonstandard, high-precision versions of FDTD to solve the wave equation and Maxwell’s equations, which is one of the main purposes of this book. We first develop analytical solutions against which numerical solutions can be compared. Topics: • FDTD algorithms • Second-order versus fourth-order models • Nonstandard finite difference models • Numerical stability • Discrete Green’s function