  We consider a gradient flow of the total variation in a negative Sobolev
space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If
$s=0$, the flow is nothing but the classical total variation flow. If $s=1$,
this is the fourth order total variation flow. We consider a convex variational
problem which gives an implicit-time discrete scheme for the flow. By a duality
based method, we give a simple numerical scheme to calculate this minimizing
problem numerically and discuss convergence of a forward-backward splitting
scheme. Several numerical experiments are given.
