  We propose and analyze an algorithm for the solution of the $L^2$-subgradient
flow of the total variation functional. The algorithm involves no
regularization, thus the numerical solution preserves the main features that
motivate practitioners to consider this type of energy. We propose an iterative
scheme for the solution of the arising problems, show that the iterations
converge, and develop a stopping criterion for them. We present numerical
experiments which illustrate the power of the method, explore the solution
behavior, and compare with regularized flows.
