We describe a hybrid method for the numerical solution of the inviscid
Burgers equation. The proposed scheme consists of
a Fourier Galerkin spectral method for the resolution of the large scales
of the solution coupled to a real space method capable of reproducing
the small scale features such as shocks. The hybrid scheme captures correctly the decay of the energy of the solution. The underlying idea of the scheme
fits in the optimal prediction framework where prior knowledge about an
equation is used to improve upon existing algorithms.