A moving spectral element method is used to investigate the effects of non-Newtonian lubricants on the dynamics of a journal bearing. It is well known that for a constant viscosity full-film lubricant the journal eventually displays half-speed whirl that is undesirable due to the accompanying fall in load bearing capacity and vibration. The inclusion of a cavitation model into the scheme stabilizes the motion of the journal in some situations. In this paper it is demonstrated that piezoviscous lubricants can stabilize the journal even without the incorporation of a cavitation model. The addition of a cavitation model is shown to stabilize the motion of the journal further in the sense that equilibrium points or limit cycles are reached more quickly. The effects of shear thinning and inertia on the motion of the journal are also considered.