4 Mono-dimensional tests

4.1 General considerations

In order to validate this modified transport algorithm, we present
some 1D tests, and we compare the results of the standard method
and of the FARGO method on a realistic test problem. We solve simultaneously
the continuity and Navier Stokes equation for an isothermal gas (which
has a non-vanishing but small kinematic viscosity):

We assume that at rest the system has a uniform density and sound speed . The waves which can propagate in this system have the following dispersion relationship:

which reduces to the standard dispersion relation for an undamped acoustic wave provided the system is evolved for a time small compared to the damping timescale . This will be the case for the results we are going to present below, so that any apparent damping of the waves has a numerical origin. We do the following:

- 1.
- We first analyze the propagation of a sound wave in the matter
frame, i.e. we take as initial conditions:

where*s*is the wave relative amplitude. The polarization adopted corresponds to a rightwards propagating wave. According to Eq. (19), it propagates with a phase velocity which is . We study this propagation with the standard transport algorithm (we are in the matter frame so there is no systematic average*x*-velocity, hence no need for a FARGO algorithm). We check that in this case the solution we get is accurate by varying the timestep and checking that the solution has converged. - 2.
- We then turn to a case where the setup is slightly modified.
We take:

where*v*_{0}is a constant, which we choose much bigger than (which would correspond to the conditions of a thin keplerian disk, for example). The evolution of the system from this setup ought to be the same as before, since it merely corresponds to the same physical situation, but described from a frame moving at a constant speed -*v*_{0}wrt the first one, so one can invoke Galilean invariance to conclude that the wave profile evolution has to be the same. So any "good'' algorithm should approach as closely as possible the results of the matter frame simulations. We show that this is not quite the case with the standard transport method, which suffers from quite a high numerical dissipation, whereas FARGO behaves much better (not to mention its much faster execution). As a side result we also show that in this problem taking a CFL effective ratio (for the standard transport method) bigger than leads to an artificial and non-linear increase of the wave profile, and hence has to be avoided.

Now if we just change
the initial velocity by uniformly adding 1.0 to them
at *t*=0, which means that we are no more in the matter frame, and
we still work with the standard transport algorithm,
then we get the dotted profile, which has
the amplitude
obtained from the computation in the matter frame. In this run the
CFL ratio is
.
In order to check the
timestep dependency of this result, we redo this test with twice
as smaller a timestep (
)
and we get
the dash-dotted profile, which has about twice as smaller a density
contrast than the previous curve. Note that if this effect were
to be due to a physical kinematic viscosity ,
then its value
should be:
,
much higher than the expected viscosity in a minimum mass protoplanetary
disk (
in our dimensionless units).
Now, instead of decreasing the timestep, we increase it and set
(hence the CFL ratio is about 0.64).
We then get at time *t*_{0} the dot-dot-dot-dashed profile, which is
not numerically damped but slightly amplified. With such a large
timestep, we can use the modified transport algorithm, which in
that case corresponds to a rightwards one cell shift and a leftwards
normal transport with a remaining CFL ratio of
1-0.64=0.36. In that case
we get the thin long-dashed profile. If we use the modified FARGO transport
algorithm, we can still increase the timestep. The thin solid
profile and the thin short-dashed profile have been obtained
respectively with
(effective CFL ratio )
and
(effective CFL ratio ).
We clearly see from these results that the FARGO transport
algorithm leads to less numerical dissipation than the standard
transport.
From the
first two tests in the non-comoving frame, one can conclude that
increasing the *number* of timesteps over a given
time interval with the standard transport algorithm
increases the numerical dissipation (if the
grid is moving wrt the matter frame with a
velocity
and if the main part of the velocity
comes from *v*_{0}). A simple explanation for the
lower numerical dissipation of the FARGO algorithm is
that
it requires less iterations as the timestep increases,
and since most of the distance swept is achieved through an
exact shift (a circular permutation), the numerical dissipation
has to decrease as the timestep increases.

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