Introduction:

The transport of charged particles plays a central role in the simulation of several models of fundamental interest for physics and engineering, in particular of semiconductor devices and plasma physics for nuclear fusion research. From the environmental point of view, the first topic is important as for saving energy and silicon, the second one as for avoiding the risks directly connected to other forms of energy production (like nuclear fission).
The transistor is the fundamental block of any electronic device. In the seventies, the size of FETs inside a CPU was roughly 10000 nm, while now the commercial size is around 30 nm. The research is focused on obtaining 10 nm SOI devices, which requires an effort for the modeling and from the computational point of view in order to provide solvers that take into account all the relevant phenomena emerging at this scale. These solvers are divided essentially into two categories: Monte Carlo and deterministic. The first ones are widely used in the engineering community thanks to their efficiency and their implementation. Among the deterministic solvers, there exist different accuracy levels: the mesoscopic ones aim at being very precise despite their computational cost, while the hydrodynamic or fluid ones have a lowerdimensional domain and are hence faster but coarser; several are the strategies applied for the timespace discretization and for the time integration in the scientific community: Strang splitting, RungeKutta, semiLagrangian schemes, Finite Differences, Discontinuous Galerkin, waterbags, etc.
A huge effort is being made in order to obtain energy from nuclear fusion. Instead of using very heavy atoms and extracting energy by bombarding them (nuclear fission), energy is extracted by fusioning light atoms. The advantages are clear in terms of safety, because there is no chain reaction to control, and environment, because no CO2 is produced, the radioactive waste products are much shorterlived and the fuels could be produced locally. Much research has been developed in the last decades; the main problem to overcome now is how to control the instabilities propagating inside the reactor chamber for a plasma dense enough so as to achieve a reasonable gain factor (Q $\thickapprox$ 10 is the goal). The best model to describe the evolution of the plasma inside the reactor chamber would be a full 3D VlasovMaxwell system, which is 6D in the phase space thus unbearable from the numerical point of view. Therefore, reduced models are taken into account and simulated.
In this special session we would like to achieve two objectives: by the one hand, sketch what the goals and the physics background are in the aforementioned fields; on the other hand, give an overview of the stateoftheart numerical strategies used by the scientific community, namely by mathematicians, engineers and physicists, for the simulation of these problems. 
