  We present a complete formulation of second-order (2+1)-dimensional
anisotropic hydrodynamics. The resulting framework generalizes leading-order
anisotropic hydrodynamics by allowing for deviations of the one-particle
distribution function from the spheroidal form assumed at leading order. We
derive complete second-order equations of motion for the additional terms in
the macroscopic currents generated by these deviations from their kinetic
definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of
coupled partial differential equations for the momentum-space anisotropy
parameter, effective temperature, the transverse components of the fluid
four-velocity, and the viscous tensor components generated by deviations of the
distribution from spheroidal form. We then perform a quantitative test of our
approach by applying it to the case of one-dimensional boost-invariant
expansion in the relaxation time approximation (RTA) in which case it is
possible to numerically solve the Boltzmann equation exactly. We demonstrate
that the second-order anisotropic hydrodynamics approach provides an excellent
approximation to the exact (0+1)-dimensional RTA solution for both small and
large values of the shear viscosity.
