Transport and diffusion (MAT/MAP 567). Fifth edition

This course presents the mathematical and numerical study of models of partial differential equations, the so called transport and diffusion equations. These equations are modelling the evolution of a density of particles or individuals in interaction with their environment. After an introduction of the main models and of their origin in nuclear reactor physics, radiative transfer or dynamics of structured population, the course presents the free transport model, i.e. in absence of collisions of particles with the ambient environment, and then the linear Boltzmann equation. The limit of the diffusion of transport models is then addressed in the case of a small mean free travel of particles. Some finite difference methods of resolution of diffusion and transport equations are then presented and discussed. The theory of critical calculation is described and interpreted as the resolution of an eigenvalue problem to determine a stationary-in-time status. Issues of homogenisation are finally addressed and the relationship is shown between transport at a microscopic scale and diffusion at a macroscopic scale.