144
3. Beyond the One-Way Wave Equation
The standard Von Neumann analysis shows that the numerical solution will be
nonamplifying when
for all k in the interval [0, n / Ar]. Here, as before, J1- = C / Sx and v =
M M / (/),.x)2. Necessary and sufficient conditions for the stability of this scheme
are
I
o s v s 2 and J12
:s 2v.
(3.82)
In order to establish the necessity of these conditions note that the first condition
is required for stability when k /),.x = n , and the second condition is required for
stability in the limit k S:»
0, in which case
In order to establish that (3.82) is sufficient for stability, suppose that J1-2 :s 2v.
Then
lAd :s (1 - 2v(I - cosk/),.x))2 + 2v sin 2 k Sx
= I - 2v(1 - 2v)(1 - cosk/),.x)2,
which is less than unity whenever 0 :s v :s !.Note that the stable region of the
V-J1- plane defined by (3.82) is a portion of a parabola, whereas the stable region
of the
plane defined by (3.76) is a cirele.
3.4.3 Advection with Sources and Sinks
Sources and sinks typically appear as functions of the temporal and spatial coordinates and the undifferentiated unknown variables. The evolution of the unknown
variables in the pure source-sink problem is therefore govemed by ordinary differential equations. Elementary numerical methods for the solution of ordinary
differential equations have been discussed in Section 2.3 and in the preceding
Section . Further details may be found in standard texts such as Gear (197 I) or
Iserles (1996). In the following we will consider the combined effects of advection and sources or sinks in two prototypical cases.
First, suppose that the source or sink is a function only of the coordinate variables, in which case the advection-source-sink equation is
-
81/1 +c- = s(x, t) .
81/1
8t
8x
Almost any finite-difference scheme suitable for the approximation of the pure
advection problem can be trivially modified to approximate the preceding equation. The only subtlety involves the numerical specification of s(x , t). It is natural
to specify s(x, t) at the finest spatial and temporal scales resolvable on the space-
3. Beyond the One-Way Wave Equation
The standard Von Neumann analysis shows that the numerical solution will be
nonamplifying when
for all k in the interval [0, n / Ar]. Here, as before, J1- = C / Sx and v =
M M / (/),.x)2. Necessary and sufficient conditions for the stability of this scheme
are
I
o s v s 2 and J12
:s 2v.
(3.82)
In order to establish the necessity of these conditions note that the first condition
is required for stability when k /),.x = n , and the second condition is required for
stability in the limit k S:»
0, in which case
In order to establish that (3.82) is sufficient for stability, suppose that J1-2 :s 2v.
Then
lAd :s (1 - 2v(I - cosk/),.x))2 + 2v sin 2 k Sx
= I - 2v(1 - 2v)(1 - cosk/),.x)2,
which is less than unity whenever 0 :s v :s !.Note that the stable region of the
V-J1- plane defined by (3.82) is a portion of a parabola, whereas the stable region
of the
plane defined by (3.76) is a cirele.
3.4.3 Advection with Sources and Sinks
Sources and sinks typically appear as functions of the temporal and spatial coordinates and the undifferentiated unknown variables. The evolution of the unknown
variables in the pure source-sink problem is therefore govemed by ordinary differential equations. Elementary numerical methods for the solution of ordinary
differential equations have been discussed in Section 2.3 and in the preceding
Section . Further details may be found in standard texts such as Gear (197 I) or
Iserles (1996). In the following we will consider the combined effects of advection and sources or sinks in two prototypical cases.
First, suppose that the source or sink is a function only of the coordinate variables, in which case the advection-source-sink equation is
-
81/1 +c- = s(x, t) .
81/1
8t
8x
Almost any finite-difference scheme suitable for the approximation of the pure
advection problem can be trivially modified to approximate the preceding equation. The only subtlety involves the numerical specification of s(x , t). It is natural
to specify s(x, t) at the finest spatial and temporal scales resolvable on the space-