  We develop a new Lagrangian approach --- flow dynamic approach to effectively
capture the interface in the Allen-Cahn type equations. The underlying
principle of this approach is the Energetic Variational Approach (EnVarA),
motivated by Rayleigh and Onsager
\cite{onsager1931reciprocal,onsager1931reciprocal2}. Its main advantage,
comparing with numerical methods in Eulerian coordinates, is that thin
interfaces can be effectively captured with few points in the Lagrangian
coordinate. We concentrate in the one-dimensional case and construct numerical
schemes for the trajectory equation in Lagrangian coordinate that obey the
variational structures, and as a consequence, are energy dissipative. Ample
numerical results are provided to show that only a fewer points are enough to
resolve very thin interfaces by using our Lagrangian approach.
