Numerical models for fusion

At very high temperature electrons leave the atom to which they are attached and a gas of charged particles called a plasma is obtained. Plasmas, because of their interaction with an electromagnetic field, have a global and rich behaviour, much more complex than for neutral gases. This makes their study complex and fascinating. Plasma physics and its lighthouse application, controlled thermonuclear fusion, thus offers new challenges to mathematicians. Two main models are generally used to describe the dynamics of a plasma : on the one hand the so-called kinetic models which describe the evolution of the phase space density of the different particle species, and on the other hand the fluid models which describe the evolution of macroscopic quantities, like density in physical space, mean velocity and temperature. This two types of models need to be coupled with Maxwell's équations in order to describe the non linear interactions with the electromagnetic field generated by the charged particles of the plasma. This book results from notes of the lectures that were given at CIRM in Luminy during the summer school of CEMRACS 2010. It détails for topics in the mathematical and 'numerical study of plasmas : asymptotic preserving (AP) numerical schemes that enable to deal consistently with two space or time scales, gyrokinetic simulations of magnetic fusion plasmas, MHD simulations of magnetic fusion plasmas and the mathematical study of Landau damping.