  In dynamical systems, it is advantageous to identify regions of flow which
can exhibit maximal influence on nearby behaviour. Hyperbolic Lagrangian
Coherent Structures have been introduced to obtain two-dimensional surfaces
which maximise repulsion or attraction in three-dimensional dynamical systems
with arbitrary time-dependence. However, the numerical method to compute them
requires obtaining derivatives associated with the system, often performed
through the approximation of divided differences, which can lead to significant
numerical error and numerical noise. In this paper, we introduce a novel method
for the numerical calculation of hyperbolic Lagrangian Coherent Structures
using Differential Algebra called DA-LCS. As a form of automatic forward
differentiation, it allows direct computation of the Taylor expansion of the
flow, its derivatives, and the eigenvectors of the associated strain tensor,
with all derivatives obtained algebraically and to machine precision. It does
so without a priori information about the system, such as variational equations
or explicit derivatives. We demonstrate that this can provide significant
improvements in the accuracy of the Lagrangian Coherent Structures identified
compared to finite-differencing methods in a series of test cases drawn from
the literature. We also show how DA-LCS uncovers additional dynamical behaviour
in a real-world example drawn from astrodynamics.
