A $1$ -harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in $\mathbb{R}^N$ , is formulated by the use of subdifferentials of a singular energythe total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result, a local-in-time solution of $1$ -harmonic map flow equation is constructed as a limit of the solutions of $p$ -harmonic $(p>1)$ map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.