Transport on adaptive random lattices

Abstract

In this paper, we present a method for the solution of those linear transport processes that may be described by a master equation, such as electron, neutron, and photon transport, and more exotic variants thereof. We base our algorithm on a Markov process on a Voronoi-Delaunay grid, a nonperiodic lattice which is derived from a random point process that is chosen to optimally represent certain properties of the medium through which the transport occurs. Our grid is locally translation and rotation invariant in the mean. We illustrate our approach by means of a particular example, in which the expectation value of the length of a grid line corresponds to the local mean free path. In this example, the lattice is a direct representation of the "free path space" of the medium. Subsequently, transport is defined as simply moving particles from one node to the next, interactions taking place at each point. We derive the statistical properties of such lattices, describe the limiting behavior, and show how interactions are incorporated as global coefficients. Two elementary linear transport problems are discussed: that of free ballistic transport, and the transport of particles through a scattering medium. We also mention a combination of these two. We discuss the efficiency of our method, showing that it is much faster than most other methods because the operation count does not scale with the number of sources. We test our method by focusing on the transport of ionizing radiation through a static medium, and show that the computed results for the classical test case of an ionization front expanding in a homogeneous medium agree perfectly with the analytic solution. We finish by illustrating the efficiency and flexibility of our method with the results of a simulation of the reionization of the large scale structure of the Universe.