Nonlinear diffusive transport and shock acceleration

Dominik Walter

We explore a nonlinear diffusive type of particle/cosmic ray transport. A special focus will be put on particles/cosmic rays, escaping from a shock or other localized acceleration sites. Instead of solving coupled differential equations, as is the more common method of describing the interaction of diffusing particles with the backgrond medium, we analyse a single nonlinear advection-diffusion equation.
In a first step we analyse the effect of the nonlinear model on particle transport, we apply different analytical techniques to cartesian and spherical symetrical geometries, to derive exact solutions wherever possible and establish numerical models to compare and expand on this results. As a succesful analytical tool we use similiarity solutions and approximations via fundamental solutions, with different strengths and limitations. As a foundation for the numerical models we use the grid based Code VLUGR3, to provide numerical solutions, when there is no analytical way of solving distinct models.
As a second step we construct a model for shock acceleration, to investigate the impact of nonlinear diffusion on shock acceleration, again using VLUGR3. We recreate a linear cartesian case of reference with a given analytical solution and study the occuring differences in the spectra and denstiy profiles under the inluence of nonlinear diffusion.