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stable for arbitrary time-step sizes—however, they can't converge to the
correct answer unless the CFL condition is met.
To further muddy the waters, there is a related quantity called the
CFL
number
, often denoted
α
.If
c
is the maximum speed of information prop-
agation in the problem—assuming this concept makes sense, for example
in the advection equation we're studying (where
c
=max
) or in certain
wave equations where it might be termed the “speed of sound”—then the
CFL number
α
of a given discretization is defined from
|
u
|
Δ
t
=
α
Δ
x
c
.
(3.4)
Thus the time step we talked about above, inequality (3.2), could be ex-
pressed as taking a CFL number of five. The CFL condition for explicit
finite difference schemes can be expressed as a limit on the CFL num-
ber; similarly the stability of some, though not all, explicit finite difference
schemes can be conveniently expressed as another limit on the CFL num-
ber. The CFL number by itself is just a useful parameter, not a condition
on anything.
3.4 Dissipation
Notice that in the interpolation step of semi-Lagrangian advection we are
taking a weighted average of values from the previous time step. That is,
with each advection step, we are doing an averaging operation. Averaging
tends to smooth out or blur sharp features, a process called
dissipation
.In
signal-processing terminology, we have a low-pass filter. A single blurring
step is pretty harmless, but if we repeatedly blur every time step, you can
imagine there are going to be problems.
Let's try to understand this smoothing behavior more physically. We'll
use a technique called
modified PDEs
. The common way of looking at
numerical error in solving equations is that our solution gets perturbed from
the true solution by some amount: we're only approximately solving the
problem. The approach that we'll now use, sometimes also called
backwards
error analysis
, instead takes the perspective that we
are
solving a problem
exactly—it's just the problem we solved isn't quite the same as the one we
started out with, i.e., the problem has been perturbed in some way. Often
interpreting the error this way, and understanding the perturbation to the
problem being solved, is extremely useful.
