Active matter is the study of many-body systems driven out of equilibrium at a local level. Typical examples are found in biology and span many length scales, such as flocks of birds or tissues of cells. This thesis focuses on non-momentum conserving (dry), polar systems, whose hydrodynamics are generically described by the Toner-Tu equations. These systems exhibit a variety of emergent behaviour, such as collective motion and phase separation, which often only emerge at high densities.

By adapting the standard lattice Boltzmann method for fluid mechanics, we develop a new method for simulating dry, polar active fluids. In particular this method is easy to implement and effective at high densities. Through a Chapman-Enskog style expansion, we confirm that the corresponding macroscopic equations are the Toner-Tu equations, and connect the system parameters with the coefficients of the equations.

We demonstrate the functionality and adaptability of our method by recreating two different phenomena: motility-induced phase separation and collective motion. Furthermore, by incorporating contact inhibition of locomotion effects into the collective motion model, we uncover two new first order phase transitions and a potentially new critical transition. We interpret these transitions through a stability analysis.

In addition, we perform a stability analysis on an open interface of a fluid that obeys an incompressible version of the Toner-Tu equations. We find that collective motion stabilises the interface, but the interface is unstable when starting from a stationary state. This has implications for both wound healing and the results of our simulations.