Direct Vlasov solvers

TL;DR

This paper presents a Hamiltonian-based framework for developing Vlasov solvers to analyze coherent instabilities in synchrotrons, emphasizing a general linearized equation form and practical implementation steps.

Contribution

It introduces a Hamiltonian formalism for Vlasov equations and details a method to construct solvers for transverse beam instabilities.

Findings

Derived a compact linearized Vlasov equation using Poisson brackets.

Outlined a procedure for building Vlasov solvers.

Applied the method to transverse instabilities from beam coupling impedance.

Abstract

In these proceedings we will describe the theory and practical steps required to build Vlasov solvers such as those commonly used to compute coherent instabilities in synchrotrons. Thanks to a Hamiltonian formalism, we will derive a compact and general form of the linearized Vlasov equation, written using Poisson brackets. This in turn will be the basis of a procedure to build Vlasov solvers, applied to the specific example of transverse instabilities arising from beam coupling impedance.