alex    >>> /Users/epolyach/Dropbox/Urtica/Gas_FEM; /Users/epolyach/Dropbox/WORK/Gaseous_modes
perseus >>> /home/Tux/epolyach/GAS_FEM/CED0

FEM for gaseous discs

Here I show some issues I bumped while calculating modes in the gaseous disc with pressure and zero softened gravity, \(Q=1\).
FEM provides “major” and “minor” modes. The major modes are labelled by letters: R, A, B, C, D, E, F, G corresponding to the decrease of the pattern speed:

I use three codes, S6, PG, and MN that differ in the weight functions \(\mu, \nu\) (see also subsection 'Codes’). For PG I use \(\mu=\nu=\Sigma_0(R)\), while S6 code has simply \(\mu=\nu=1\).

Minor modes issue – N_d dependence

For N_d=4, minor modes attain growth rate 0.0225 (number of modes is N_m=25) for \(R_\textrm{max}=16\), and 0.009 (N_m=61) for \(R_\textrm{max}=32\). This is different from what I have in the version I of the paper [0.0176 (76) and 0.076 (33)].

My experiments show that the maximum growth rate increases and their number decreses when one switches from N_d=1 to N_d=4. In particular, for N_d=2, max growth rate is 0.0085, the number of modes is 67 (although, there are garbage modes with relatively high growth rates, so I might miss one or two minor modes).

Eigenfunctions are different:

Numerical instability

Green points of minor modes are localized both to the left and to the right from the boundary \(\Omega_\textrm{p}=0.462\) separating emitters from absorbers:

Examination of the eigenfunctions shows that this is a result of numerical instability due to insufficient number of nodes.

Here is a typical regular eigenfunction \((\Omega_\textrm{p}, \omega_\textrm{I})=(0.738, 0.020)\):

Here is an effect of numerical instability \((\Omega_\textrm{p}, \omega_\textrm{I})=(0.738, 0.016)\) – filled areas are oscillations of the functions:

Grid

Comparison of lin-log grid and linear grid; S6 code

Codes

@ ~/Dropbox/Urtica/Gas_FEM:

gas_FEM_PG_Rmax_perseus.m

solves matrix equation for \(u\), \(v\), \(\eta\); weights \(\mu=\nu=(1+\tau\Sigma_0(R))\), \(\tau\) is usually taken \(10^8\).

@ perseus : /home/Tux/epolyach/GAS_FEM/CED0

gas_FEM_S6_Rmax_perseus.m

solves matrix equation for \(u\), \(v\), \(\eta\); weights \(\mu=\nu=1\).

@ ~/Dropbox/Urtica/Gas_FEM:

gas_FEM_MuNu_Rmax.m

solves matrix equation for \(u\), \(v\), \(\eta\); weights \(\mu\), \(\nu\) are arbitrary analytic functions.

Grids

I use grids:

• linear (\(R_\textrm{min}\), \(R_\textrm{max}\), \(N+1\))

• log (\(R_\textrm{min}\), \(R_\textrm{max}\), \(N+1\))

• linear–log: \(R_j = R_s (e^{q \tau_j} - 1)\).

Matlab usage

perseus : Intel(R) Xeon(R) CPU E5-2667 v2 @ 3.30GHz 8 cores, RAM 128 Gb
edita   : Intel(R) Xeon(R) CPU E3-1240 v5 @ 3.50GHz 4 cores, RAM 32 Gb
MBP     : Intel(R) Core(TM) i7-7820HQ CPU @ 2.90GHz 4 cores, RAM 16 Gb