  Anisotropic hydrodynamics improves upon standard dissipative fluid dynamics
by treating certain large dissipative corrections non-perturbatively.
Relativistic heavy-ion collisions feature two such large dissipative effects:
(i) Strongly anisotropic expansion generates a large shear stress component
which manifests itself in very different longitudinal and transverse pressures,
especially at early times. (ii) Critical fluctuations near the quark-hadron
phase transition lead to a large bulk viscous pressure on the conversion
surface between hydrodynamics and a microscopic hadronic cascade description of
the final collision stage. We present a new dissipative hydrodynamic
formulation for non-conformal fluids where both of these effects are treated
nonperturbatively. The evolution equations are derived from the Boltzmann
equation in the 14-moment approximation, using an expansion around an
anisotropic leading-order distribution function with two momentum-space
deformation parameters, accounting for the longitudinal and transverse
pressures. To obtain their evolution we impose generalized Landau matching
conditions for the longitudinal and transverse pressures. We describe an
approximate anisotropic equation of state that relates the anisotropy
parameters with the macroscopic pressures. Residual shear stresses are smaller
and are treated perturbatively, as in standard second-order dissipative fluid
dynamics. The resulting optimized viscous anisotropic hydrodynamic evolution
equations are derived in 3+1 dimensions and tested in a (0+1)-dimensional
Bjorken expansion, using a state-of-the-art lattice equation of state.
Comparisons with other viscous hydrodynamical frameworks are presented.
