3.4 Diffusion, Sourees, and Sinks
141
since
= cosI (ci) - iX),
the condition for absolute stability" reduces to
x= 0 and Ici)I < 1.
The region of the X-ci) plane within which the leapfrog scheme is stable is just a
line segment along the ci)-axis. Any amount of diffusion destabilizes the leapfrog
scheme.
A stable approximation to (3.75) that uses leapfrog time differencing for the
osciIIatory forcing may be obtained by evaluating the diffusion term at time level
n - I such that
A,n+1
A,n-I
= iwif>n + )..if>n-I.
(3.77)
'I'
- 'I'
211t
ci)2 - 1 _
The region of the X-ci) plane that satisfies this inequality lies inside the curve la -
beled LFF in Fig. 3.7a. Consideration of the case 1 + 2X ci)2 shows that the
actual region of the X -ci) plane throughout which IAufl
1 is a larger triangular
region with vertices at (X, ci)) equal to (0, 1), (-1 ,0), and (0, -I). This larger
region of formal stability is of no practical use, however, since Alff is pure imaginary when 1 + 2X ci)2, and as a consequence the numerical solution undergoes
Note that although the standard leapfrog method is accurate to 0 [(l1t)2] , this approach introduces an O(l1t) truncation error in the approximation ofthe diffusion
term. The amplification factor for the leapfrog-forward scheme (3.77) is
Alff = ici) ± (_ci)2 + I + 2X)I/2.
When 1 + 2X > ci)2, the square root is real ; both amplification factors have the
same magnitude, and
IAufl = 1 + 2)...
2
-
Xs 0, or
It follows that the leapfrog-forward sc heme will be stable when 1 + 2X > ci)2 and
- - - 2
-
a 411t oscillation independent of the actual value of wl1t . Because the region of
useful stability for the leapfrog-forward scheme is relatively smalI, this scheme
is appropriate only for problems with very low viscosity. Even when the value of
/..l1t is as small as 0.3, stability considerations require a significant reduction in
wl1t relative to that which would be stable in the inviscid limit.
Much better stability properties can be obtained using A-stable finite-difference
schemes. An A-stable finite-difference approximation to (3.75) is absolutely stable for all )..ßt
O. An A-stable method generates a bounded numerical solu4 As discussed in Section 2.3.4, this scheme is subject to a weak instability when iiJ = ± 1 and
):= 0.
141
since
= cosI (ci) - iX),
the condition for absolute stability" reduces to
x= 0 and Ici)I < 1.
The region of the X-ci) plane within which the leapfrog scheme is stable is just a
line segment along the ci)-axis. Any amount of diffusion destabilizes the leapfrog
scheme.
A stable approximation to (3.75) that uses leapfrog time differencing for the
osciIIatory forcing may be obtained by evaluating the diffusion term at time level
n - I such that
A,n+1
A,n-I
= iwif>n + )..if>n-I.
(3.77)
'I'
- 'I'
211t
ci)2 - 1 _
The region of the X-ci) plane that satisfies this inequality lies inside the curve la -
beled LFF in Fig. 3.7a. Consideration of the case 1 + 2X ci)2 shows that the
actual region of the X -ci) plane throughout which IAufl
1 is a larger triangular
region with vertices at (X, ci)) equal to (0, 1), (-1 ,0), and (0, -I). This larger
region of formal stability is of no practical use, however, since Alff is pure imaginary when 1 + 2X ci)2, and as a consequence the numerical solution undergoes
Note that although the standard leapfrog method is accurate to 0 [(l1t)2] , this approach introduces an O(l1t) truncation error in the approximation ofthe diffusion
term. The amplification factor for the leapfrog-forward scheme (3.77) is
Alff = ici) ± (_ci)2 + I + 2X)I/2.
When 1 + 2X > ci)2, the square root is real ; both amplification factors have the
same magnitude, and
IAufl = 1 + 2)...
2
-
Xs 0, or
It follows that the leapfrog-forward sc heme will be stable when 1 + 2X > ci)2 and
- - - 2
-
a 411t oscillation independent of the actual value of wl1t . Because the region of
useful stability for the leapfrog-forward scheme is relatively smalI, this scheme
is appropriate only for problems with very low viscosity. Even when the value of
/..l1t is as small as 0.3, stability considerations require a significant reduction in
wl1t relative to that which would be stable in the inviscid limit.
Much better stability properties can be obtained using A-stable finite-difference
schemes. An A-stable finite-difference approximation to (3.75) is absolutely stable for all )..ßt
O. An A-stable method generates a bounded numerical solu4 As discussed in Section 2.3.4, this scheme is subject to a weak instability when iiJ = ± 1 and
):= 0.