Development of a Vlasov Solver for Arbitrary Sub-optimal Lengthening Conditions in Double-RF System

Solving Vlasov equation is a classic method for analyzing collective beam instabilities. Considering longitudinal impedance and the nonlinear longitudinal potential well, we developed a new Vlasov solver which can be used to study the transverse mode-coupling instability under the arbitrary sub-optimal lengthening and the optimal lengthening conditions in a double-RF system. Several different techniques to deal with the radial direction of longitudinal phase space have been tested. Numerical discretization method is selected in this paper. The development of the solver is presented in details here. Benchmarks and crosscheck of the solver have been made and presented as well.