  Discrete flow mapping was recently introduced as an efficient ray based
method determining wave energy distributions in complex built up structures.
Wave energy densities are transported along ray trajectories through polygonal
mesh elements using a finite dimensional approximation of a ray transfer
operator. In this way the method can be viewed as a smoothed ray tracing method
defined over meshed surfaces. Many applications require the resolution of wave
energy distributions in three-dimensional domains, such as in room acoustics,
underwater acoustics and for electromagnetic cavity problems. In this work we
extend discrete flow mapping to three-dimensional domains by propagating wave
energy densities through tetrahedral meshes. The geometric simplicity of the
tetrahedral mesh elements is utilised to efficiently compute the ray transfer
operator using a mixture of analytic and spectrally accurate numerical
integration. The important issue of how to choose a suitable basis
approximation in phase space whilst maintaining a reasonable computational cost
is addressed via low order local approximations on tetrahedral faces in the
position coordinate and high order orthogonal polynomial expansions in momentum
space.
