1
EE256 Numerical Electromagnetics
H. O. #6
Marshall
24 June 2008
Summer 2008
LECTURE 3
3.1
FINITE DIFFERENCE SOLUTIONS OF THE CONVECTION EQUATION
As the simplest example of a PDE, we consider ﬁnite difference solutions of the convection equation [2.20].
As mentioned before, this equation constitutes the simplest prototype for hyperbolic equations, and is also
highly relevant for our FDTD analyses of Maxwell’s equations, since it has the same form as the component
curl equations that we shall solve in FDTD solutions of electromagnetic problems. Consider the convection
equation written in terms of voltage on a transmission line:
∂
V
∂t
+
v
p
∂
V
∂z
= 0
[3.1]
where
v
p
’
1
ft(ns)

1
, i.e., the speed of light in free space. As mentioned in Lecture #2, the exact
solution of this convection equation is:
V
(
z,t
) =
V
+
(
z

v
p
t
)
[3.2]
where
V
+
(
·
)
is an arbitrary function. A particular solution is determined by means of the initial condition
at time
t
= 0
, i.e.,
V
+
(
z,
0) = Φ(
z
)
, where
Φ(
z
)
is the initial voltage distribution on the line. The solution
at any later time is then given by:
V
(
z,t
) = Φ(
z

v
p
t
)
[3.3]
In other words, the initial voltage distribution simply is convected (or propagated) to the right at the
velocity
v
p
, preserving its magnitude and shape.
We shall now consider FDTD solutions of [3.1] for a particular initial voltage distribution:
V
(
z,
0) = Φ(
z
) =
12
z
V
0
≤
z
≤
0
.
5ft
,

12(1

z
)
V
0
.
5ft
≤
z
≤
1
.
0ft
,
0
V
Elsewhere
[3.4]
The exact solution is simply the triangular pulse moving to the right at the speed of
v
p
’
1
ft(ns)

1
, as
shown in Figure 3.1.
We now proceed to numerically solve for
V
(
z,t
)
, using two different methods, namely (i) the forward
time centeredspace method, and (ii) the leapfrog method.
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0.5
1
1.5
2
2.5
2
4
6
8
t
=0
v
p
t
=0.25 ns
t
=0.75 ns
z
(ft)
V
(
z,t
)
(V)
Figure 3.1:
Exact solution of the convection equation for a triangular initial voltage
distribution.
The triangular distribution simply moves to the right at the speed
v
p
’
1
ft(ns)

1
.
3.1.1
The ForwardTime CenteredSpace Method
As our ﬁrst approximation of the PDE given in [3.1], we use the ﬁrstorder forwarddifference approximation
(i.e., equation [2.42]) for the time derivative and the secondorder centereddifference approximation (i.e.,
equation [2.55]) for the space derivative. We thus have:
∂
V
∂t
+
v
p
∂
V
∂z
= 0
→
V
n
+1
i
V
n
i
Δ
t
+
v
p
V
n
i
+1
V
n
i

1
2Δ
z
= 0
[3.5]
We now solve [3.5] for
V
n
+1
i
to ﬁnd:
Forwardtime
centeredspace
V
n
+1
i
=
V
n
i

±
v
p
Δ
t
2Δ
z
²
³
V
n
i
+1
V
n
i

1
´
[3.6]
In comparing different ﬁnite difference methods, it is useful to graphically depict the grid points involved
in the evaluation of the time and space derivatives. Such a mesh diagram is given in Figure 3.2, for both of
the methods used in this section.
For the solution of [3.6], we choose
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 Derivative, Electromagnet, NS, finite difference, Finite difference method, Finite differences

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