Collisional Particle Disks
  We present a new, simple, fast algorithm to numerically evolve disks of inelastically colliding particles surrounding a central star. Our algorithm adds negligible computational cost to the fastest existing collisionless N-body codes, and can be used to simulate, for the first time, the interaction of planets with disks over many viscous times. Though the algorithm is implemented in two dimensions-i.e., the motions of bodies need only be tracked in a plane-it captures the behavior of fully three-dimensional disks in which collisions maintain inclinations that are comparable to random eccentricities. We subject the algorithm to a battery of tests for the case of an isolated, narrow, circular ring. Numerical simulations agree with analytic theory with regards to how particles' random velocities equilibrate; how the ring viscously spreads; and how energy dissipation, angular momentum transport, and material transport are connected. We derive and measure the critical value of the coefficient of restitution above which viscous stirring dominates inelastic damping and the particles' velocity dispersion runs away. 
Yoram Lithwick, Eugene Chiang