Up: FARGO: A fast eulerian
Subsections
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):
 |
(17) |
 |
(18) |
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:
 |
(19) |
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:
 |
(20) |
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:
 |
(21) |
where v0 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 -v0 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.
We deal with a
1D grid composed of
cells, with periodic boundary conditions.
The cell width is
=0.0314,
the isothermal sound speed is
.
The equilibrium density is
.
These parameters
correspond roughly to the ones used in the numerical study of
a protoplanet on a circular orbit at 5 AU
embedded in a minimum mass protoplanetary disk
(Hayashi et al.1985 or Bryden et al.1998),
that are described in Sect. 5),
when the central star mass and the protoplanet orbit radius are taken
to be respectively
the units of mass and distance. We present
the results of different test runs
in Fig. 1. The thick solid line represents
the initial profile, which corresponds to a rightward propagating
acoustic wave, with wavelength
.
The relative amplitude of this
sound wave is s=10-2. The thick dashed line represents
the density profile at time t0=220, i.e. after the wave has
traveled
times its own wavelength, when studied
in the matter frame, i.e. when the velocity at t=0 is set to
be only the perturbed velocity associated to the sound wave. The
thick dashed profile is obtained with the standard transport
algorithm (there is no need for the modified one in this
case since we work in the matter frame), with a timestep
.
The curves obtained by choosing a
much smaller timestep appear to coincide exactly with this one,
hence we can consider this thick dashed line as the actual state
the system must have at the date t0. This profile does not
exactly coincide with the initial one because t0 is
of the profile steepening time
.
 |
Figure 1:
Compared evolution of an acoustic wave evolved
with the standard
transport algorithm and with the modified transport algorithm. We
plot only two of the five wavelengths, i.e. 80 cells out of 200.
Due to numerical effects the phase velocity of all these profiles
do not exactly coincide with ,
so that after a time t0 their
phases do not coincide.
For this reason the profiles
have been shifted so that they have all approximately the same phase
in order to improve the clarity of the plot |
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 t0 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 v0). 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.
Up: FARGO: A fast eulerian
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