3.4 Diffusion, Sourees, and Sinks
139
by numerical diffusion unless the molecular diffusivity is very large or the grid
resolution is very fine. In order to accurately represent the diffusion term in low -
viscosity flow, it is generally necessary to use a less diffusive approximation to
01/1lax, such as a centered difference or a higher-order one-sided difference.
As the horizontal resolution increases, the numerical Peclet number decreases,
and in principle, there is some grid size at which diffusive transport dominates
advective transport in the shortest resolvable modes. Nevertheless, in many problems involving low-viscosity flow this grid size may be several orders of magnitude smaller than the physical scales of primary interest, so that there is no possibility of resolving the scales at which molecular viscosity dominates numerical
diffusion without exceeding the resources of the most advanced computers. Even
when molecular diffusion has no direct influence on the resolved-scale fields, an
essentially inviscid transport by sub-grid-scale eddies may produce turbulent mixing whose influence on the resolved-scale fields is often parametrized by a diffu -
sion term in which the true molecular diffusivity M is replaced by an "eddy"
diffusivity M (Yih 1977, p. 572). The eddy diffusivity is generally parametrized
such that M is proportional to the mesh size, in which case the numerical Peclet
number c IM does not decrease as the grid is refined, and it is not possible to
make the parametrized eddy diffusion dominate the numerical diffusion by reducing the mesh size.
Now consider the effects of time-differencing on the stability of finite -difference
approximations to the advection-diffusion equation. In order to isolate the role of
time-differencing on the numerical solution, (3.72) is Fourier transformed with
respect to the x coordinate to obtain
dt
d1/J
2 A
+ ick1/l = - M k 1/1.
A
The coefficients in the preceding ordinary differential equation can be further
simplified to yield the following prototype equation for the investigation of timedifferences in the advection-diffusion problem:
d1/l
dt
-
= I(lJ1/I + ) . . ,1/1,
.
(3.75)
where wand)", are real. Solutions to (3.75) satisfy 11/I(t)1
11/1(0)1, provided
that A
O. Practically useful numerical approximations to (3.75) must satisfy
the analogous stability condition that ln 1 1<1>°1 . Numerical approximations to
(3.75) computed with a specific value ofthe parameters
and
are defined
to be absolutely stable if ln 1 1<1>°, for all n .
The amplification factor associated with a forward-difference approximation to
(3.75) is
Ar = 1 +i+iw,
where w=
and i =
Thus the forward-difference approximation to
(3.75) will be absolutely stable when
(1 + i)2 + w 2 s 1.
(3.76)
139
by numerical diffusion unless the molecular diffusivity is very large or the grid
resolution is very fine. In order to accurately represent the diffusion term in low -
viscosity flow, it is generally necessary to use a less diffusive approximation to
01/1lax, such as a centered difference or a higher-order one-sided difference.
As the horizontal resolution increases, the numerical Peclet number decreases,
and in principle, there is some grid size at which diffusive transport dominates
advective transport in the shortest resolvable modes. Nevertheless, in many problems involving low-viscosity flow this grid size may be several orders of magnitude smaller than the physical scales of primary interest, so that there is no possibility of resolving the scales at which molecular viscosity dominates numerical
diffusion without exceeding the resources of the most advanced computers. Even
when molecular diffusion has no direct influence on the resolved-scale fields, an
essentially inviscid transport by sub-grid-scale eddies may produce turbulent mixing whose influence on the resolved-scale fields is often parametrized by a diffu -
sion term in which the true molecular diffusivity M is replaced by an "eddy"
diffusivity M (Yih 1977, p. 572). The eddy diffusivity is generally parametrized
such that M is proportional to the mesh size, in which case the numerical Peclet
number c IM does not decrease as the grid is refined, and it is not possible to
make the parametrized eddy diffusion dominate the numerical diffusion by reducing the mesh size.
Now consider the effects of time-differencing on the stability of finite -difference
approximations to the advection-diffusion equation. In order to isolate the role of
time-differencing on the numerical solution, (3.72) is Fourier transformed with
respect to the x coordinate to obtain
dt
d1/J
2 A
+ ick1/l = - M k 1/1.
A
The coefficients in the preceding ordinary differential equation can be further
simplified to yield the following prototype equation for the investigation of timedifferences in the advection-diffusion problem:
d1/l
dt
-
= I(lJ1/I + ) . . ,1/1,
.
(3.75)
where wand)", are real. Solutions to (3.75) satisfy 11/I(t)1
11/1(0)1, provided
that A
O. Practically useful numerical approximations to (3.75) must satisfy
the analogous stability condition that ln 1 1<1>°1 . Numerical approximations to
(3.75) computed with a specific value ofthe parameters
and
are defined
to be absolutely stable if ln 1 1<1>°, for all n .
The amplification factor associated with a forward-difference approximation to
(3.75) is
Ar = 1 +i+iw,
where w=
and i =
Thus the forward-difference approximation to
(3.75) will be absolutely stable when
(1 + i)2 + w 2 s 1.
(3.76)