Graphics Reference
In-Depth Information
To make our analysis as simple as possible, we'll solve the advection
problem in one dimension with a constant velocity
u>
0:
∂q
∂t
+
u
∂q
∂x
=0
.
We'll assume Δ
t<
Δ
x/u
, i.e., that the particle trajectories span less than
a grid cell—the analysis easily extends to larger time steps too, but noth-
ing significant changes. In that case, the starting point of the trajectory
that ends on grid point
i
is in the interval [
x
i−
1
,x
i
].
Doing the linear
interpolation between
q
i−
1
and
q
i
at point
x
i
−
Δ
tu
gives
q
i−
1
+
1
q
i
.
Δ
tu
Δ
x
Δ
tu
Δ
x
q
n
+1
i
=
−
We can rearrange this to get
Δ
tu
q
i
−
q
i−
1
Δ
x
q
n
+1
i
=
q
i
−
,
(3.5)
which is in fact exactly the Eulerian scheme of forward Euler in time and a
one-sided finite difference in space.
4
Now recall the Taylor series for
q
i−
1
:
∂q
∂x
n
Δ
x
+
∂
2
q
∂x
2
n
Δ
x
2
2
q
i−
1
=
q
i
−
+
O
(Δ
x
3
)
.
i
i
Substituting this into Equation (3.5) and doing the cancellation gives
∂q
∂x
n
∂
2
q
∂x
2
n
+
O
(Δ
x
3
)
Δ
x
2
2
1
Δ
x
q
n
+1
i
=
q
i
−
Δ
tu
Δ
x
−
i
i
Δ
tu
∂q
∂x
n
+Δ
tu
Δ
x
∂
2
q
∂x
2
n
=
q
i
−
+
O
(Δ
x
2
)
.
i
i
Up to a second-order truncation error, we can see this is forward Euler in
time applied to the
modified PDE
:
∂t
+
u
∂q
∂x
=
u
Δ
x
∂
2
q
∂q
∂x
2
.
This is the advection equation with an additional viscosity-like term with
coecient
u
Δ
x
! (Recall from the momentum equation of Navier-Stokes
4
If you're interested, note that the side to which the finite difference is biased is the
side from which the fluid is flowing. This is no coincidence and makes perfect physical
sense—in the real physical world, you get information from upwind, not downwind,
directions in advection. In general, biasing a finite difference to the direction that flow
is coming from is called
upwinding
. Most advanced Eulerian schemes are upwind-biased
schemes that do this with more accurate finite difference formulas.
