Geography Reference
In-Depth Information
advection equation, the original differential equation system has solutions whose
amplitudes remain constant in time. In the example (13.8), stability considerations
place stringent limitations on the value of the parameter σ , as we show later.
If the initial condition is specified as
Re
exp (ikx)
=
=
q (x, 0)
cos (kx)
The analytic solution of (13.6) that satisfies this initial condition is
q (x, t)
=
Re
{
exp [ik (x
−
ct)]
} =
cos (kx
−
ct)
(13.10)
We now compare (13.10) with the solution of the finite difference system (13.8)
and (13.9). In finite difference form the initial condition is
q
m,0
=
ˆ
exp (ikmδx)
=
exp (ipm)
(13.11)
where p
kδx. Noting that the analytic solution (13.10) is separable in x and t ,
we consider solutions of (13.8) and (13.9) of the form
≡
B
s
exp (ipm)
q
m,s
=
ˆ
(13.12)
where B is a complex constant. Substituting into (13.8) and dividing through by
the common factor B
s
−
1
, we obtain a quadratic equation in B:
B
2
+
(2i sin θ
P
) B
−
1
=
0
(13.13)
where sin θ
p
≡
σ sinp. Equation (13.3.3) has two roots, which may be expressed
in the form
iθ
p
The general solution of the finite difference equation is thus
exp
−
iθ
p
,B
2
=−
exp
+
B
1
=
q
m,s
=
CB
1
+
DB
2
exp (ipm)
Ce
i
(
pm
−
θ
p
s
)
1)
s
e
i
(
pm
+
θ
p
s
)
ˆ
=
+
D (
−
(13.14)
where C and D are constants to be determined by the initial conditions (13.11) and
the first time step (13.9). The former gives C
+
D
=
1, whereas the latter gives
Ce
−
iθ
p
De
+
iθ
p
−
=
1
−
i sin θ
p
(13.15)
Thus,
1
cos θ
p
2 cos θ
p
1
cos θ
p
2 cos θ
p
+
−
C
=
,D
=−
(13.16)
