Comparative study of forward and backward test-kinetic simulation approaches

In this paper we perform a comparative study of the forward and backward Liouville mapping applied to the modeling of ring-shaped and non-gyrotropic velocity distribution functions of particles injected in a sheared electromagnetic field. The test-kinetic method is used to compute the velocity distribution function in various areas of a proton cloud moving in the vicinity of a region with a sharp transition of the magnetic field and a non-uniform electric field. In the forward approach the velocity distribution function is computed for a two-dimensional spatial bin, while in the backward approach the distribution function is averaged over a spatial bin with the same size as for the forward method and using a two-dimensional trapezoidal integration scheme. It is shown that the two approaches lead to similar results for spatial bins where the velocity distribution function varies smoothly. On the other hand, with bins covering regions of configuration space characterized by sharp spatial gradients of the velocity distribution function, the forward and backward approaches will generally provide different results.