  We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson
scattering in $P$-wave, by calculating correlation functions using lattice QCD
with light quark masses such that $m_\pi = 236$ MeV in a cubic volume of $\sim
(4 \,\mathrm{fm})^3$. Variational analyses of large matrices of correlation
functions computed using operator constructions resembling $\pi\pi$,
$K\overline{K}$ and $q\bar{q}$, in several moving frames and several lattice
irreducible representations, leads to discrete energy spectra from which
scattering amplitudes are extracted. In the elastic $\pi\pi$ scattering region
we obtain a detailed energy-dependence for the phase-shift, corresponding to a
$\rho$ resonance, and we extend the analysis into the coupled-channel
$K\overline{K}$ region for the first time, finding a small coupling between the
channels.
