  We introduce and analyze a spectral vanishing viscosity approximation of
periodic fractional conservation laws. The fractional part of these equations
can be a fractional Laplacian or other non-local operators that are generators
of pure jump L\'{e}vy processes. To accommodate for shock solutions, we first
extend to the periodic setting the Kru\v{z}kov-Alibaud entropy formulation and
prove well-posedness. Then we introduce the numerical method, which is a
non-linear Fourier Galerkin method with an additional spectral viscosity term.
This type of approximation was first introduced by Tadmor for pure conservation
laws. We prove that this {\em non-monotone} method converges to the entropy
solution of the problem, that it retains the spectral accuracy of the Fourier
method, and that it diagonalizes the fractional term reducing dramatically the
computational cost induced by this term. We also derive a robust $L^1$-error
estimate, and provide numerical experiments for the fractional Burgers'
equation.
