We formulate a lattice Boltzmann model which simulates compressible fluids. By choosing the parameters of the equilibrium distribution appropriately, we are able to select the sound speed (which may be set arbitrarily low), bulk viscosity, and kinematic viscosity. This model simulates compressible flows and can include shocks. With a proper rescaling and zero-sound speed, this model simulates Burgers's equation. The viscosity determined by a Chapman-Enskog expansion compares well with that measured from simulations. We also compare the exact solutions of Burgers's equation on the unit circle to solutions of our lattice Boltzmann model, again finding reasonable agreement.