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.