  The $I=1$ $p$-wave and $I=2$ $s$-wave elastic $\pi$-$\pi$ scattering
amplitudes are calculated from a first-principles lattice QCD simulation using
a single ensemble of gauge field configurations with $N_{\mathrm{f}} = 2+1$
dynamical flavors of anisotropic clover-improved Wilson fermions. This ensemble
has a large spatial volume $V=(3.7\mathrm{fm})^3$, pion mass $m_{\pi} =
230\mathrm{MeV}$, and spatial lattice spacing $a_s = 0.11\mathrm{fm}$.
Calculation of the necessary temporal correlation matrices is efficiently
performed using the stochastic LapH method, while the large volume enables an
improved energy resolution compared to previous work. For this single ensemble
we obtain $m_{\rho}/m_{\pi} = 3.350(24)$, $g_{\rho\pi\pi} = 5.99(26)$, and a
clear signal for the $I=2$ $s$-wave. The success of the stochastic LapH method
in this proof-of-principle large-volume calculation paves the way for
quantitative study of the lattice spacing effects and quark mass dependence of
scattering amplitudes using state-of-the-art ensembles.
