The

Finite Difference Time Domain (

FDTD) method is a technique for solving time-dependent

vector partial differential equations for which there may be no analytic solutions.

FDTD is used extensively to solve boundary-value problems in

mechanical,

fluid and

electromagnetic analyses.

The FDTD is at its heart an implementation of finite differencing. Finite differencing was first developed in the 1920's for solving nonlinear hydrodynamics equations. To implement the FDTD, the analyst first selects a finite region which is to be discretized. The region is then gridded or meshed into uniform cells whose size is generally less than some pre-defined amount. Then, the differential equations governing the physics of the problem are replaced by finite difference equations. This reduces the equation to an algebraic one, which can then be solved numerically using a time-step or space-stepping (or both) scheme which is subject to the initial and boundary conditions.

Recall from calculus the definition of the derivative:

lim(Δ x → 0) f'(x) = (f(x + Δ x) - f(x)) / (Δx)
The **forward difference** approximation around the point Xo would be

f'(Xo) ≅ (f(Xo + Δ x) - f(Xo)) / (Δ x)
We might also use the **backward difference** approximation:

f'(Xo) ≅ (f(Xo) - f(Xo - Δ x)) / (Δ x)
Or the **central difference** approximation :

f'(Xo) ≅ (f(Xo + Δ x) - f(Xo - Δ x))/ (2 Δ x)
The second derivative can be similarly obtained by applying the first-order differencing twice.

FDTD is attractive because it may be used to solve problems in which the domain and the solutions contain nonlinearities. It is also possible to simulate the response of a system to specific excitation waveforms which may not be possible using frequency domain techniques.

Among the drawbacks of the FDTD method is the computer memory required to store the grid and vector field quantities at each grid point. In three-dimensional
simulations which may encompass a large simulation domain, this requirement can grow to be quite large. Also, many simulations require the introduction of absorbing boundaries to simulate semi-infinite space around the region. These ficticious absorbers are not perfect and can add to inaccuracy in the solution.

A good starting reference in the FDTD technique is "Computational Electrodynamics : The Finite-Difference Time-Domain Method", by Allen Taflove and Susan C. Hagness, Artech House, June 2000.

References : Jianming Jin, Computational Electromagnetics course notes, ECE Department, The University of Illinois at Urbana-Champaign, May 1998.