d’Alembert Digitized: A Wave Pulse Derivation of the Finite Difference Time Domain Method for Numerically Solving Maxwell’s Equations

An alternative way of deriving the Finite Difference Time Domain method (FDTD) for simulating the dynamics of electromagnetic waves in matter for one dimensional systems with grid spacing and material properties that vary with position is presented. This alternative derivation provides useful insight into the physics of electromagnetic waves in matter and the properties of FDTD. The method uses d'Alembert's splitting of waves into forward and backward pulses of arbitrary shape to solve the one-dimensional wave equation. Constant velocity of waves in dispersionless dielectric materials, partial reflection and transmission at boundaries between materials with different indices of refraction, and partial reflection, transmission, and attenuation through conducting materials are derived in the process of deriving the method so that real physics is learned simultaneously with the numerics. The traditional FDTD equations are reproduced from a model of wave pulses partially reflecting, transmitting, and attenuating through regions of constant current. The alternative method therefore shows the physical effects of the finite difference approximation in a way that is easy to visualize. The method allows for easy derivation of some results that are more complicated to derive with the traditional method, such as showing that FDTD is exact for dielectrics when the ratio of the spatial and temporal grid spacing is the wave speed, incorporating reflectionless boundary conditions, and showing the method retains second order accuracy when the grid spacing varies with position and the material parameters make sudden jumps across layer boundaries.