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2.1 Explicit 2 levels
Slide : [ 2 levels
Scheme 
Code 
Run 
Stability 
Upwind 
VIDEO
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A spatial difference to the left for the advection (1.3.1#eq.1)
and a centered difference for the diffusion (1.3.2#eq.1) for
example yields an explicit scheme for the advectiondiffusion equation
that involves only two time levels
Figure 2.1#fig.1:
Explicit 2 levels.

The CourantFriedrichsLewy (CFL) number
and the coefficient
measure typical advection and
diffusion velocities relative to the characteristic propagation
speed of the mesh
.
For every step in time, a new value of the solution fp[j] is
obtained explicitly from a linear combination involving the nearest
neighbors f[j1],f[j],f[j+1] , which are known either from a
previous step or the initial condition. This scheme has been
implemented in JBONE
as
for (int j=1; j<n; j++) {
fp[j]=f[j] beta *(f[j]f[j1])+alpha*(f[j+1]2.*f[j]+f[j1]); }
fp[0]=f[0] beta *(f[0]f[ n ])+alpha*(f[ 1 ]2.*f[0]+f[ n ]);
fp[n]=f[n] beta *(f[n]f[n1])+alpha*(f[ 0 ]2.*f[n]+f[n1]);
where the last two statements take care of the periodicity.
Sharp variations of the solution should be AVOIDED to produce
physically meaningful results; to study the numerical properties of
a approximation, it is however often illuminating to initialize a
box function
and check how the intrinsic numerical dispersion / damping of a
scheme affects the superposition of both the short and the long
wavelengths.
The example below
shows the result of an evolution with a constant advection
and
no physical diffusion
.
JBONE applet: press Start/Stop
to simulate the advection of a box function and to test how the
explicit 2 levels scheme affects the dispersion and the damping
of short and long wavelengths superposed in a box function.

After 128 steps of duration
, the pulse propagates exactly
once accross the periodic domain, which is here discretized with 64 mesh
points homogeneously distributed over the length
so that
.
The lowest order
moment
(density, (1.2.5#eq.1)) is conserved to a very good accuracy
(as can be judged from the JBONE monitor under Mom[%] )
and the function remains positive everywhere as it should(as can be judged from Min )
. The shape, however, is strongly affected as the sharp edges are smoothed
out due intrinsic numerical damping.
The properties of a FD approximation are best understood from a
numerical dispersion / Von Neuman stability analysis, which
is carried out on a homogeneous grid
using a
hamonic ansatz
.
Define the amplification factor as
and simplify by
to cast the dispersion relation into
Figure 2.1#fig.2 illustrates with vectors in the complex plane
how the first three term are combined in the presence of advection only
(
).
Figure 2.1#fig.2:
Numerical dispersion / stability (explicit 2 levels).

Short wavelengths
are heavily damped
(area inside the outer circle) if the CFL number is smaller than unity
; for values exceeding the magic time step where
,
the short wavelengths however grow into a socalled numerical
instability.
Reload
the initial applet
parameters and try to reproduce this in JBONE.
For a diffusive process (
), the dispersion relation
(2.1#eq.3) shows that the 2 levels scheme is stable for
all wavelengths provided that
,
i.e.
.
Although the superposition of advection and diffusion is slightly more
limiting, the overall conditions for numerical stability can conveniently
be summarized as
It is finally worth pointing out that a backward finite difference for
the advection is unstable for negative velocities
; this defect of
noncentered schemes is sometimes cured with an upwind difference,
which takes the finite difference forward or backward depending on the
local direction of propagation (exercise 2.01).
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Next: 2.2 Explicit 3 levels
