  Ray flow methods are an efficient tool to estimate vibro-acoustic or
electromagnetic energy transport in complex domains at high-frequencies. Here,
a Petrov-Galerkin discretization of a phase-space boundary integral equation
for transporting wave energy densities on two-dimensional surfaces is proposed.
The directional dependence of the energy density is approximated at each point
on the boundary in terms of a finite local set of directions propagating into
the domain. The direction of propagation can be preserved for transport across
multi-component domains when the directions within the local set are inherited
from a global direction set. The range of applicability and computational cost
of the method will be explored through a series of numerical experiments,
including wave problems from both acoustics and elasticity in both single and
multi-component domains. The domain geometries considered range from both
regular and irregular polygons to curved surfaces, including a cast aluminium
shock tower from a Range Rover car.
