  We present the first direct numerical simulation of gravitational wave
turbulence. General relativity equations are solved numerically in a periodic
box with a diagonal metric tensor depending on two space coordinates only,
$g_{ij} \equiv g_{ii}(x,y,t) \delta_{ij}$, and with an additional small-scale
dissipative term. We limit ourselves to weak gravitational waves and to a
freely decaying turbulence. We find that an initial metric excitation at
intermediate wavenumber leads to a dual cascade of energy and wave action. When
the direct energy cascade reaches the dissipative scales, a transition is
observed in the temporal evolution of energy from a plateau to a power-law
decay, while the inverse cascade front continues to propagate toward low
wavenumbers. The wavenumber and frequency-wavenumber spectra are found to be
compatible with the theory of weak wave turbulence and the characteristic
time-scale of the dual cascade is that expected for four-wave resonant
interactions. The simulation reveals that an initially weak gravitational wave
turbulence tends to become strong as the inverse cascade of wave action
progresses with a selective amplification of the fluctuations $g_{11}$ and
$g_{22}$.
