  We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method
for solving the three-dimensional (3D) incompressible Euler equations. This
method evolves advected quantities by discretizing the flow map associated with
the velocity field. Using the properties of the Lie group of volume preserving
diffeomorphisms SDiff, long-time deformations are computed from a composition
of short-time submaps which can be accurately evolved on coarse grids. This
method is a fundamental extension to the CM method for two-dimensional
incompressible Euler equations [51]. We take a geometric approach in the 3D
case where the vorticity is not a scalar advected quantity, but can be computed
as a differential 2-form through the pullback of the initial condition by the
characteristic map. This formulation is based on the Kelvin circulation theorem
and gives point-wise a Lagrangian description of the vorticity field. We
demonstrate through numerical experiments the validity of the method and show
that energy is not dissipated through artificial viscosity and small scales of
the solution are preserved. We provide error estimates and numerical
convergence tests showing that the method is globally third-order accurate.
