1 Finite-difference time-domain
(FDTD) basics
The finite-difference time-domain (FDTD) method is a powerful
numerical approach to solve Maxwell(‘)s equations in the time-domain using
discretization of space and time. This article is a first-principles
overview of the algorithm.
2 Theory
Maxwell(‘)s equations in the
time-domain in free space are written as
[begin{eqnarray}
label{maxwell1}nablatimesmathbf{E}&=&frac{-partialmathbf{B}}{partial
t}:
(Faraday’s;law)\
label{maxwell2}nablatimesmathbf{H}&=&mathbf{J}+frac{partialmathbf{D}}
{partial t}:(Ampere’s;law)\
label{maxwell3}nablacdotmathbf{D}&=&rho\
label{maxwell4}nablacdotmathbf{B}&=&0
end{eqnarray}]
where (mathbf{E}), (mathbf{H}), (mathbf{D}), (mathbf{B}), (rho) and (mathbf{J}) denote electric field,
magnetic field, electric flux density, magnetic flux density, external
charge density, and electric current density, respectively. Maxwell(‘)s equations can be expanded in three
dimensions (3D) as
[begin{eqnarray}
label{expanded1}frac{partial E_{x}}{partial
t}=frac{1}{varepsilon_{0}}left
(frac{partial H_{z}}{partial y}-frac{partial H_{y}}{partial
z}right)\
label{expanded2}frac{partial E_{y}}{partial
t}=frac{1}{varepsilon_{0}}left
(frac{partial H_
{x}}{partial z}-frac{partial H_{z}}{partial x}right)\
label{expanded3}frac{partial E_{z}}{partial
t}=frac{1}{varepsilon_{0}}left
(frac{partial H_
{y}}{partial x}-frac{partial H_{x}}{partial y}right)\
label{expanded4}frac{partial H_{x}}{partial
t}=frac{1}{mu_{0}}left(frac
{partial E_{y}}
{partial z}-frac{partial E_{z}}{partial y}right)\
label{expanded5}frac{partial H_{y}}{partial
t}=frac{1}{mu_{0}}left(frac
{partial E_{z}}
{partial x}-frac{partial E_{x}}{partial z}right)\
label{expanded6}frac{partial H_{z}}{partial
t}=frac{1}{mu_{0}}left(frac
{partial E_{x}}
{partial y}-frac{partial E_{y}}{partial x}right)
end{eqnarray}]
where the electric and magnetic fields are related to the
corresponding flux densities through the freespace values of the
electric permitivity ((varepsilon_{0})) and magnetic
permeability ((mu_{0})) as
[begin{eqnarray}
mathbf{D}&=&varepsilon_{0}mathbf{E}\
mathbf{B}&=&mu_{0}mathbf{H}
end{eqnarray}]
3 Discretization in space and
time
As can be seen the expanded Maxwell equations include continuous
derivatives in time and space which are required to be solved
numerically. These derivatives are approximated by central differences
with (2^ {nd}) order accuracy
[begin{equation}
label{central_diff}frac{partial{f(x)}}{partial{x}}approxfrac{f(x+Delta
x)-f
(x-Delta x)}
{Delta x}
end{equation}]
Figure 1 provides a visual representation of the central finite
difference in comparison with the exact definition of the first
derivative. In order to support the central differences and have the
relationship between (bf{E}) and
(bf {H}) it is necessary to offset
the two fields. This discretization is accomplished using Yee cells in a
rectangle grid [1-2], see figure 2. These Yee cells form meshing in the
grid to sample the electric and magnetic fields. Each individual
component of one field is surrounded by the four components of the other
field due to the special arrangement of the field components.
In further support of the relationship between (bf{E}) and (bf{H}), the updating process is performed
using leapfrog scheme [1-2] in which the electric and magnetic field
components are evaluated alternately in the time domain such that the
updated electric field components are obtained using the updated
magnetic field components that were calculated in the preceding step.
Figure 3 illustrates this process for 1D sequential updating the
electric and magnetic field.
The central difference expression, along with the leapfrogging in
time allows the approximation of Maxwell(‘)s equations to obtain update
equations. For instance, in the two-dimensional case with the transverse
electric (TE: (H_x), (H_z), and (E_y) ) polarization is obtained through
the 2D-Yee scheme (the xz-cut of the 3D-Yee cell including (H_x), (H_z), and (E_y)). The
discretized field components can be derived from the expanded equations
for (H_x), (H_z), and (E_y) when
(frac{partial}{partial
y}rightarrow0) as
[begin{eqnarray}
E_{y}^{n+1}(i,k) & = & E_{y}^{n}(i,k)+frac{Delta
t}{varepsilon_{0},Delta
z}left[H_{x}^{n+0.5}(i,k)-H_{x}^{n+0.5}(i,k-1)right]nonumber \
& & -frac{Delta t}{varepsilon_{0},Delta
x}left[H_{z}^{n+0.5}(i,k)-H_{z}^
{n+0.5}(i-1,k)right]\
H_{x}^{n+0.5}(i,k)&=&H_{x}^{n-0.5}(i,k)+frac{Delta
t}{mu_{0},Delta z}left[E_
{y}^{n}(i,k+1)-E_{y}^{n}(i,k)right]\
H_{z}^{n+0.5}(i,k)&=&H_{z}^{n-0.5}(i,k)-frac{Delta
t}{mu_{0},Delta x}left[E_
{y}^{n}(i+1,k)-E_{y}^{n}(i,k)right]
end{eqnarray}]
where the superscript n represents the time step. The indices i and k
denote the spatial indices along the x and z directions. The discretized
equations for the transverse magnetic (TM: (E_x), (E_z), and (H_y)) can also be similarly obtained in
the same manner.