Code validation
A.2 Boltzmann solver
The hydrodynamical part of the code is validated in the previous section. In this section, the Boltzmann solver employed in chapter 3 is validated. They are described in Sumiyoshi
& Yamada (2012), Nagakura et al. (2014), and Nagakura et al. (2017).
A.2.1 Validations with the static background
The basic tests for the Boltzmann solver implemented in the Boltzmann-radiation-hydrodynamics code with the static background are reported in Sumiyoshi & Yamada (2012). In the following, code validations in the paper are described in order to convince the reader that the code successfully works. The tests conducted in the following are the
Figure A.2. The steady-state solution of the rotational equilibrium of the stellar matter.
The density profiles of the initial condition (exact, red solid lines) and at t = 100 s from the beginning of calculation (Nr grids, green dashed lines) along the rotational axis are shown. The employed radial grid numbers are Nr = 230 (upper panel) and Nr = 460 (lower panel). This figure is reproduced from Nagakura et al. (2011) by permission of the AAS.
spatial diffusion of neutrinos in the optically thick regime, the free-streaming of neutrinos in the optically thin regime, the energy spectrum of the steady-state neutrinos, the time evolution to the equilibrium state, the absorptivity and emissivity of neutrinos, and the steady-state neutrino distributions with realistic background of the CCSN in 1D and 2D.
First, the diffusion of neutrinos is tested. In the optically thick limit, the neutrinos behave diffusively. Therefore the analytic solution of the diffusing Gaussian packet is useful. It is expressed as
f(r, t) =f0
( t0
t0+t )α
exp {
− |r−r0|2 4D(t0+t)
}
, (A.1)
where r0, f0, t0, D = cλ/3, c, and λ are the central position of the packet, the initial central value of the distribution, the initial time, the diffusion coefficient, the light speed, and the mean free path of the isotropic scattering, respectively. The indexα = Nd/2 is related to the spatial dimension, Nd. Note that the time t is measured from the initial timet0.
6x10-3 5 4 3 2 1 0 neutrino density [fm-3 ]
5x105 4
3 2
1 0
radius [cm]
t = 0 s
t = 1x10-4 s
Figure A.3. The spherical diffusion of the Gaussian packet of neutrinos. The radial pro-files of the numerical solutions of the neutrino densities at t = 0 s (blue crosses) and t= 10−4s (red crosses) are compared to the analytic solution (black solid lines). This figure is reproduced from Sumiyoshi & Yamada (2012) by permission of the AAS.
First, in figure A.3, the diffusion of the packet at the coordinate center is shown. Since r0 = 0, the solution is spherically symmetric although the calculation is done in 3D.
The computational domain is 0≤ r ≤ 5 km, 0 ≤ θ ≤ π, and 0 ≤ ϕ ≤ 2π, respectively, and the grid numbers are (Nr, Nθ, Nϕ, Nϵ, Nθν, Nϕν) = (80,18,36,2,12,12). The mean free path is set to λ = 103cm and the initial time is set to t0 = 2.5×10−4s so that the initial width of the packet is 1 km. Figure A.3 implies that the spherical diffusion of the Gaussian packet is correctly solved by the Boltzmann solver since the numerical and analytic solutions coincide.
Next, the diffusion of the 2D Gaussian packet is shown in figure A.4. The center of the packet r0 is located at 1,000 km on the equator. The computational domain is a square with a side of 10 km whose center is the same as the packet. Since the domain is located far from the center, the coordinates are almost the Cartesian coordinates. It is useful to employ the coordinates spanned by Z = rcosθ and R = rsinθ instead of r and θ for the presentation in figure A.4. The grid numbers are (Nr, Nθ, Nϵ, Nθν, Nϕν) = (100,96,4,12,12). Again,λ= 103cm and t0= 2.5×10−4s. From figure A.4, it is proven that the Boltzmann solver correctly treats the 2D diffusion.
Then, the Boltzmann solver in the optically thin limit is tested. In the optically thin region, the neutrinos freely stream with the light speed c. In order to realize the free-streaming, the neutrino reactions are switched off. In figure A.5, the 1D
6x10-3 5 4 3 2 1 0 neutrino density [fm-3 ]
1.0040x108 1.0000
0.9960
R [cm]
t = 5.0x10-4 s t = 0 s
Figure A.4. The 2D diffusion of the Gaussian packet of neutrinos. The numerical so-lutions (crosses) of the neutrino densities at t = 0 s and t = 10−4s are compared to the analytic solution (red and blue solid lines). The profile is along Z =−8.1 km. This figure is reproduced from Sumiyoshi & Yamada (2012) by permission of the AAS.
advection of the step-like-distributed neutrinos is shown. The grid numbers are (Nr, Nθ, Nϕ, Nϵ, Nθν, Nϕν) = (100,3,3,4,6,6). The initial condition is a step-like distri-bution as shown in the top panel of figure A.5. Here, only neutrinos with cosθν = 0.93247, which correspond to the most forward grid point in the momentum space, have the distribution shown in the figure. The azimuthal distribution is uniform. The middle and bottom panels indicate that the neutrinos freely stream with the light velocity projected onto the radial direction, ccosθν. The smearing of the step-like distribution is also seen, but it is not so problematic since the neutrino distributions inside the shock of the CCSN is not forward-peaked. The smearing is reduced when the spatial and temporal resolutions are improved though it is not shown.
The collision term is validated under the spherical symmetry subsequently. For the test, a snapshot of the CCSN whose progenitor mass is 15M⊙ at 100 ms after the core bounce simulated by the spherically symmetric code (Sumiyoshi et al., 2005) is taken as a background flow. The radial profiles of the density, temperature, and neutrino chemical potential of this snapshot are shown in figure A.6. In the following, this background is fixed and the neutrino distribution only is evolved. Firstly, the steady-state solution of the Boltzmann equation with the collision term is tested, and secondly, the time evolu-tion toward the equilibrium state is examined. For these calculaevolu-tions, the computaevolu-tional domain is 0≤ r ≤ 1.4×103km, 0 ≤θ ≤π/2, and 0 ≤ϕ ≤ π/2. The neutrino energy ranges up to 300 MeV, and the grid number isNϵ = 14.
For the steady-state solution, the numerically obtained energy spectrum of neutrinos is compared to the spectrum obtained from the formal solution. The well-known radiative
6x10-3 5 4 3 2 1 0 neutrino density [fm-3 ]
1.004x108 1.002
1.000 0.998
0.996
R [cm]
t = 0 s
6x10-3 5 4 3 2 1 0 neutrino density [fm-3 ]
1.004x108 1.002
1.000 0.998
0.996
R [cm]
t = 1.0x10-5 s
6x10-3 5 4 3 2 1 0 neutrino density [fm-3 ]
1.004x108 1.002
1.000 0.998
0.996
R [cm]
t = 2.0x10-5 s
Figure A.5. The free-streaming of neutrinos. The top panel shows the initial condition.
The red lines in the middle (t= 1.0×10−5s) and bottom (t= 2.0×10−5s) panels show the subsequent evolutions. The blue dashed lines in the two pan-els represent the wavefronts analytically obtained. This figure is reproduced from Sumiyoshi & Yamada (2012) by permission of the AAS.