Diffusion theory via asymptotics

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Abstract
It is known that classic diffusion theory can lead to negative scalar fluxes if the external source of particles is aniso-tropic. The usual derivation of diffusion theory, via a truncated spherical harmonics expansion, gives no consistent way of dealing with this lack of positivity. In this paper we show that a derivation of the diffusion equation using asymptotic methods leads to several options, in an asymptotically correct way, of ensuring positivity in the diffusion approximation. These same ideas can be used to produce alternate diffusion theories which should be more accurate than classic diffusion theory for diffusive problems. A specific such modified diffusion theory is proposed and tested.

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