3.4 Diffusion, Sources, and Sinks
143
tory term is replaced by the third-order Adams-Bashforth method. Before examining the combined Adarns-Bashforth-trapezoidal scherne, consider the simpler
third-order Adams-Bashforth approximation to the entire advection-diffusion
problem
l/Jn+l
- - - - - =
l/Jn (iW+A) ( 23€/Jn -16€/Jn-1 + 5€/Jn-2
) .
(3.80)
l:i.t
12
The region of the i-w plane for which solutions obtained using (3.80) are absolutely stable can be determined by numerically solving a cubic equation for the
amplification factor. The stable region for this third-order method, which is plotted
in Fig. 3.7b, is only slightly smaller than that for the first-order leapfrog-forward
scheme shown in Fig. 3.7a. Also shown in Fig. 3.7b is the region of absolute stability for the second-order Adams-Bashforth solution (2.46). One might suppose
that the second-order Adams-Bashforth method would be a more economical alternative to the third-order scheme. If the flow is sufficiently viscous, this can
be the case. However, as shown in Fig. 3.7b, the second-order Adams-Bashforth
method does not generate absolutely stable solutions to the undamped (A = 0)
problem. On the other hand, when A = 0, the third-order Adams-Bashforth
method is absolutely stable for Iwl:i.tI < 0.72.
If the damping term in (3.80) is approximated using trapezoidal time-differencing such that
(3.81)
the region of absolute stability expands, as indicated in Fig. 3.7b, so that it is
stable solutions to the inviscid problem, then the full advection-diffusion problem
determined almost entirely by the value of Iw l:i.t l. If l:i.t is small enough to yield
will be stable for almost all values of A < O. Therefore, the Adams-Bashforthtrapezoidal approximation appears to be the best scheme for simulating advectiondiffusion in situations where there are both regions of zero viscosity and patches
of high diffusivity. Patches of high eddy diffusivity may appear in a nominally inviscid fluid in localized regions where the flow is dynamically unstable to smallscale perturbations. The high eddy diffusion in these isolated regions can have a
severe impact on the time step of the overall numerical integration unless an artificial cap is imposed on the maximum eddy diffusivity or the time-differencing
is approximated with a very stable method Iike the Adams-Bashforth-trapezoidal
scheme.
The preceding analyses of the prototype ordinary differential equation (3.75)
are sufficient to characterize the stability of the various time differencing schemes
qualitatively, but they do not yield the precise stability limits of the complete
finite -difference equation obtained when the spatial derivatives in the advectiondiffusion equation are approximated by finite differences. As an example, consider the forward-time centered-space approximation
A,n+l A,n
'l'j
- 'l'j
l:i.t
n
2 n
+ clhx€/Jj = Möxl/Jj ·
143
tory term is replaced by the third-order Adams-Bashforth method. Before examining the combined Adarns-Bashforth-trapezoidal scherne, consider the simpler
third-order Adams-Bashforth approximation to the entire advection-diffusion
problem
l/Jn+l
- - - - - =
l/Jn (iW+A) ( 23€/Jn -16€/Jn-1 + 5€/Jn-2
) .
(3.80)
l:i.t
12
The region of the i-w plane for which solutions obtained using (3.80) are absolutely stable can be determined by numerically solving a cubic equation for the
amplification factor. The stable region for this third-order method, which is plotted
in Fig. 3.7b, is only slightly smaller than that for the first-order leapfrog-forward
scheme shown in Fig. 3.7a. Also shown in Fig. 3.7b is the region of absolute stability for the second-order Adams-Bashforth solution (2.46). One might suppose
that the second-order Adams-Bashforth method would be a more economical alternative to the third-order scheme. If the flow is sufficiently viscous, this can
be the case. However, as shown in Fig. 3.7b, the second-order Adams-Bashforth
method does not generate absolutely stable solutions to the undamped (A = 0)
problem. On the other hand, when A = 0, the third-order Adams-Bashforth
method is absolutely stable for Iwl:i.tI < 0.72.
If the damping term in (3.80) is approximated using trapezoidal time-differencing such that
(3.81)
the region of absolute stability expands, as indicated in Fig. 3.7b, so that it is
stable solutions to the inviscid problem, then the full advection-diffusion problem
determined almost entirely by the value of Iw l:i.t l. If l:i.t is small enough to yield
will be stable for almost all values of A < O. Therefore, the Adams-Bashforthtrapezoidal approximation appears to be the best scheme for simulating advectiondiffusion in situations where there are both regions of zero viscosity and patches
of high diffusivity. Patches of high eddy diffusivity may appear in a nominally inviscid fluid in localized regions where the flow is dynamically unstable to smallscale perturbations. The high eddy diffusion in these isolated regions can have a
severe impact on the time step of the overall numerical integration unless an artificial cap is imposed on the maximum eddy diffusivity or the time-differencing
is approximated with a very stable method Iike the Adams-Bashforth-trapezoidal
scheme.
The preceding analyses of the prototype ordinary differential equation (3.75)
are sufficient to characterize the stability of the various time differencing schemes
qualitatively, but they do not yield the precise stability limits of the complete
finite -difference equation obtained when the spatial derivatives in the advectiondiffusion equation are approximated by finite differences. As an example, consider the forward-time centered-space approximation
A,n+l A,n
'l'j
- 'l'j
l:i.t
n
2 n
+ clhx€/Jj = Möxl/Jj ·