Eulerian Approach to Solve the Vlasov Equation and Hybrid-Vlasov Simulations

In this chapter we give an overview on Eulerian Vlasov algorithms and their applications, including few basic and advanced examples. In fact, thanks to recent advances in high-performance computing, the applications of Vlasov codes have experienced a substantial increase. These applications range from the astrophysical to the laboratory framework through space plasmas. Eulerian Vlasov algorithms are particularly suited for investigating specific kinetic processes because of their very low noise level, especially in the presence of a strong nonlinear dynamics. This is the case, for instance, of nonlinear Landau damping and saturation of electromagnetic instabilities, for which we report to basic examples. Finally, we provide also a brief overview of their advanced applications in the context of kinetic plasma turbulence.