any perturbations. The number of cells used is $N_R=64$, $N_\phi = 1024$, and $N_z=64$. Figure \[fig:torus\_ic\] compares the radial profiles of the dimensionless density $\widehat{\rho}\equiv\rho/\rho_c$ and pressure $\widehat{p}=\alpha\widehat{\rho}$ in the equatorial plane as functions of $\widehat{R}$, at $t/T_{\rm orb}=0$, 1, and 5, where $T_{\rm orb}=2\pi/\Omega_0$. The density and pressure profiles are almost unchanged over time except near the contact discontinuity between the ring and external medium, demonstrating that the initial steady configuration is well maintained over many orbital periods.

By performing a linear-stability analysis, @kim16 found that the above equilibrium is gravitationally unstable to non-axisymmetric perturbations for a range of the azimuthal mode number $m$. The most unstable modes were found to have $m=9$ and $10$, with an almost equal growth rate ${\rm Im}(\omega)\approx 0.8(G\rho_c)^{1/2}\approx 17\,T_{\rm orb}^{-1}$ and a phase speed ${\rm Re}(\omega)/m\approx 0.3(G\rho_c)^{1/2}$, corresponding to overstability. To check if our Poisson solver can pick up these unstable modes and capture their growth, we generate random density perturbations with amplitude $10^{-5}$ and apply them to the equilibrium density $\rho$. We then evolve the system and monitor how various modes grow. Figure \[fig:gi\] plots the azimuthal profiles of the dimensionless density at $\widehat{R}=0.91$ and $\widehat{z}=0$ at a few selected epochs as well as the surface density $\Sigma=\int \rho\,dz$ at $t(G\rho_c)^{1/2}=20.2$. It is apparent that the perturbations grow as they propagate along the $\phi$-direction. In the highly nonlinear stage, the density distribution is dominated equally by the $m=9$ and $m=10$ modes, indicating that these are two fastest growing modes of the instability.

To measure the growth rate and the phase speed of each mode in the numerical simulation, we calculate the Fourier transform ${\cal L}_me^{-i\vartheta_m} \equiv \int \bar{\rho}(\phi) e^{-im\phi}\,d\phi  $ of the radia