Vlasov Solvers and Macroparticle Simulations

We review here two essential methods to evaluate growth rates of transverse coherent instabilities arising from beam-coupling impedance in a synchrotron, namely Vlasov equation solvers and tracking simulation of macroparticles. We will discuss the basics of the two methods, reviewing in particular the theoretical grounding of Vlasov solvers -- giving all the necessary formulas in the case of the DELPHI solver. We will then assess the advantages and limitations of the two methods, by showing a number of practical applications, both in hadrons machines such as the CERN LHC and SPS, or in lepton synchrotrons such as LEP. In particular, we will show how the Vlasov solver DELPHI can help understanding the relative lack of success in trying to stabilize the transverse mode coupling instability using a reactive or resistive transverse feedback in LEP.