Illustration of diffusion and equipartitioning as local processes: A numerical study using the scalar radiative transfer equation

Abstract

We study the transition from ballistic to diffusive to equipartitioned waves in scattering media using the acoustic radiative transfer equation. To solve this equation, we first transform it into an integral equation for the specific intensity and then construct a time stepping algorithm with which we evolve the specific intensity numerically in time. We handle the advection of energy analytically at the computational grid points and use numerical interpolation to deal with advection terms that do not lie on the grid points. This approach allows us to reduce the numerical dispersion, compared to standard numerical techniques. With this algorithm, we are able to model various initial conditions for the intensity field, non-isotropic scattering, and uniform scatterer density. We test this algorithm for an isotropic initial condition, isotropic scattering, and uniform scattering density, and find good agreement with analytical solutions. We compare our numerical solutions to known two-dimensional diffusion approximations and find good agreement. We use this algorithm to numerically investigate the transition from ballistic to diffusive to equipartitioned wave propagation over space and time, for two different initial conditions. The first one corresponds to an isotropic Gaussian distribution in space and the second one to a plane wave segment. We find that diffusion and equipartitioning must be treated as local rather than global concepts. This local behavior of equipartitioning has implications for Green's functions reconstruction, which is of interest in acoustics and seismology.