We present N -body simulations of unstable spiral modes in a dynamically cool collisionless disc .
We show that spiral modes grow in a thin collisionless disk in accordance with the analytical perturbation theory .
We use the particle-mesh code Superbox with nested grids to follow the evolution of unstable spirals that emerge from an unstable equilibrium state .
We use a large number of particles ( up to N = 40 \times 10 ^ { 6 } ) and high-resolution spatial grids in our simulations ( 128 ^ { 3 } cells ) .
These allow us to trace the dynamics of the unstable spiral modes until their wave amplitudes are saturated due to nonlinear effects .
In general , the results of our simulations are in agreement with the analytical predictions .
The growth rate and the pattern speed of the most unstable bar-mode measured in N -body simulations agree with the linear analysis .
However the parameters of secondary unstable modes are in lesser agreement because of the still limited resolution of our simulations .