Ecole doctorale de Physique de la région Parisienne (ED107)

4.4 Test and calibration of the code 119
near versions of NSE and EE. The modes that are the most interesting for GW are m = 2
modes and in the following, we will always choose this kind of initial data. Nevertheless,
the same calibration tests for the other linear l = m r-modes were done and gave the same
power law relations between δE and the number of steps per oscillation. Finally, we also
E
verified that the error in the phase of the modes is proportional to the square of the number
of steps per period. The relative errors in the phase for the m = 2 mode with 100, 150
and 200 steps per oscillation were, respectively, 1.64 × 10−3 , 7.33 × 10−4 and 4.12 × 10−4 .
It is easy to verify that the logarithms of these numbers are on a straight line with a slope
very close to 2.
We now shall see how we tried to improve the conservation of energy. For the free rmode,
the energy is expected to be exactly conserved. Furthermore, we know our numerical
error in the velocity is of the order of ∆t 2 . Remember that it comes from the second order
scheme used to calculate source terms :
S j+1/2 = 3Sj − Sj−1 . (4.18)
2
The basic idea is thus to modify this quantity, in the coefficients space, with an additional
term of the order of ∆t 2 , in such a way to retrieve the conservation of the energy
without changing the error in the velocity. We then modified the Equation (4.18) and
took
S j+1/2 = (3 + ε)Sj − (1 + ε)S j−1
2
(4.19)
where ε is of the order of ∆t 2 . It is calculated using values of the energy at the last two
instants and in such a way to “impose” on the energy to be conserved. Obviously, when
there are forces such as viscosity or RR, their power has to be taken into account in
the energy balance equation. In Figure 4.5 are the same curves as in Figure 4.2 but in
logarithmic scales. We added the result of this “improved conservation of energy” for the
run with 100 steps per oscillation. As the correction is local in time (done at almost each
time step), it leads to a time independent error that should be able to remain the same
as long as one could wish. This calculation took 320 s of CPU time.
4.4.2 Modes driven to instability in the rigidly rotating case
After trying to improve the conservation of energy for the free case, we did tests with
an RR force included. For instance, we switched on the RR force in order to drive toward
instability the linear m = 2 r-mode. As it has already been implied in Section 4.2.3,
we did it by using a modified version of the formulas (4.4) (see this section). For these
calculations, we compared numerical results with analytical calculations, both reached
with the same approximation. We took for the background star a homogeneous NS with
M = 1.4M⊙, R = 10 km and Ω ∼ (2π) 500 Hz. In Figure 4.6 appear two time evolutions
of the ratio between the energy and the initial energy for 100 oscillations of the mode. The
first calculation was done without the enhanced conservation of energy and the second