Transport: Non-diffusive, flux conservative initial value problems and ...

Transport equations 69
concentration
3.5
3
2.5
2
1.5
1
0.5
t=100
t=0
0 2 4 6 8 10
distance
Figure 5.7: Evolution of gaussian initial condition (amplitude 3, width 1., 257 grid points)
using an upwind differencing donor-cell algorithm α = 0.5. After 10 passes around the
grid (t = 100) the numerical diffusion has reduced the initial condition to less than half of
its amplitude and broadened the peak significantly.
is the implicit numerical diffusivity. One obvious approach (to quote Smolarkiewicz)
“is to make the advection step using a [donor cell method] and then reverse the effect
of the diffusion equation
∂c
∂t
�
∂
=
∂x
κimpl
�
∂c
∂x
(5.5.20)
in the next corrective step.
The problem is that the diffusion process and the equation that describes it are
irreversible. But it is not true that the solution of the diffusion equation cannot be
reversed in time. Just as a film showing the diffusion process may be reversed in
time, the equivalent numerical trick may be found to produce the same effect. It is
enough to notice that (5.5.20) may be written in the form
where
Vd =
�
∂c
∂t
= −∂Vdc
∂x
∂c
∂x c > 0
− κimpl
c
0 c = 0
(5.5.21)
(5.5.22)
[this scheme assumes that the advected quantity is always positive]. Here Vd will
be referred to as the “diffusion velocity.” Now, defining an “anti-diffusion velocity”
˜V =
�
−Vd c > 0
0 c = 0
(5.5.23)