  The terminal wall velocity of a first-order phase transition bubble can be
calculated from a set of fluid equations describing the scalar fields and the
plasma's state. We rederive these equations from the energy-momentum tensor
conservation and the Boltzmann equation, without linearizing in the background
temperature and fluid velocity. The resulting equations have a finite solution
for any wall velocity. We propose a spectral method to integrate the Boltzmann
equation, which is simple, efficient and accurate. As an example, we apply this
new methodology to the singlet scalar extension of the standard model. We find
that all solutions are naturally categorized as deflagrations ($v_w\sim c_s$)
or ultrarelativistic detonations ($\gamma_w\gtrsim10$). Furthermore, the
contributions from out-of-equilibrium effects are, most of the time,
subdominant. Finally, we use these results to propose several approximation
schemes with increasing levels of complexity and accuracy. They can be used to
considerably simplify the methodology while correctly describing the
qualitative behavior of the bubble wall.
