We propose efficient Eulerian methods for approximating the
finite-time Lyapunov exponent (FTLE). The
idea is to compute the related flow map using the level set method and
the Liouville equation. There are
several advantages of the proposed approach. Unlike the usual
Lagrangian-type computations, the resulting
method requires the velocity field defined only at discrete locations.
No interpolation of the velocity field
is needed. Also, the method automatically stops a particle trajectory
in the case when the ray hits the
boundary of the computational domain. The computational complexity of
the algorithm is O(1/x^(d+1))
with d the dimension of the physical space. Since there are the same
number of mesh points in the x-t space,
the computational complexity of the proposed Eulerian approach is
optimal in the sense that each grid
point is visited for only O(1) time. We also extend the algorithm to
compute the FTLE on a co-dimension
one manifold. The resulting algorithm does not require computation on
any local coordinate system and is
simple to implement even for an evolving manifold.