$p (\vec{r})$ | = | pressure or energy density of the cosmic rays |
$\stackrel{\leftrightarrow}{\kappa} $ | = | the diffusion tensor |
$\vec{v}$ | = | total velocity $\, -\vec{v}_{sw} - \gamma \vec{v}_{dr}$ |
$ a_0 $ | = | coefficient $\, -\gamma (\nabla\cdot \vec{v}_{sw})$ |
There are several factors which make the CR equation unmanageble by available
packages, despite of the fact that Convection-Diffusion equations like this are
quite common in for example Fluid Dynamics. First there is the symmetric tensor
$\kappa$, which as such could have been accounted for by a Structural Mechanics
program [ OC ].
But in our case this tensor has to be combined with convection terms.
These convection terms, together with diffusion, could have been handled by any
common Computational Fluid Dynamics (CFD) code [ SV ].
But to my knowledge there exists no CFD code which implements
a full tensor in its diffusion modelling. Moreover, due to the geometry of the
problem, the density decreases with distance like $r^{-2}$ resulting - with the
condition of mass conservation - into a velocity field with non-vanishing
divergence. It is essentially the latter feature which also gives rise to a
multiplicative $a_0$ term; this complicates the CR model even further.
The above is not to say that something very special would be needed in order to
solve the problem. On the contrary. We shall see that very conventional Finite
Element and Finite Volume techniques are quite sufficient. The important thing,
however, is that they must be used in combination. This is the basic idea
I have been advertizing several times in the past, and will continue to do so in
future : Unified Numerical Analysis .
A two-dimensional analogue of the CR equation has been studied at the Mathematics
Stack Exchange forum [ MSE ] . Which may serve as
an introduction to the more difficult 3-D problem.