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In-Depth Information
This has the advantage of always being positive, even when the initial
velocities are zero, so we avoid a divide-by-zero in inequality (3.2).
In some cases artifacts will still be present with a time step of this
size; one possible remedy that avoids the expense of running the entire
simulation at a smaller time step is to just trace the trajectories used in
semi-Lagrangian advection with several small substeps. If each substep is
limited to
Δ
t<
Δ
x
, i.e., so that each substep only traverses roughly
one grid cell, there is little opportunity for problems to arise. Note that
this substep restriction can be taken locally: in fast-moving regions of the
fluid more substeps might be used than in slow-moving regions.
|
u
(
x
)
|
3.3.1 The CFL Condition
Before leaving the subject of time-step sizes, let's take a closer look at some-
thing called the
CFL condition
. There is some confusion in the literature
about what exactly this condition is, so in this section I'll try to set the
story straight. This section can be safely skipped if you're not interested
in some more technical aspects of numerical analysis.
The CFL condition, named for applied mathematicians R. Courant,
K. Friedrichs, and H. Lewy, is a simple and very intuitive necessary condi-
tion for convergence. Convergence means that if you repeat a simulation
with smaller and smaller Δ
t
and Δ
x
, in the limit going to zero, then your
numerical solutions should approach the exact solution.
2
The solution
q
(
x
,t
) of a time-dependent partial differential equation,
such as the advection equation, at a particular point in space
x
and time
t
depends on some or all of the initial conditions. That is, if we modify
the initial conditions at some locations and solve the problem again, it will
change
q
(
x
,t
); at other locations the modifications may have no effect.
In the case of the constant-velocity advection equation, the value
q
(
x
,t
)
is exactly equal to
q
(
x
t
u,
0), so it only depends on a single point in
the initial conditions. For other PDEs, such as the heat-diffusion equation
∂q/∂t
=
−
q
, each point of the solution depends on
all
points in the
initial conditions. The
domain of dependence
for a point is precisely the set
of locations that have an effect on the value of the solution at that point.
∇·∇
2
As an aside, this is a sticky point for the three-dimensional incompressible Navier-
Stokes equations, and at the time of this writing nobody has managed to prove that
they do in fact have a unique solution for all time. It has already been proven in two
dimensions, and in three dimensions up to some finite time; a million-dollar prize has
been offered from the Clay Institute for the first person to finish the proof in three
dimensions.
