Simulation of the beam-beam effects ine+e−storage rings with a method of reduced region of mesh

Abstract

A highly accurate self-consistent particle code to simulate the beam-beam collision in e 1 e 2 storage rings has been developed. It adopts a method of solving the Poisson equation with an open boundary. The method consists of two steps: assigning the potential on a finite boundary using Green's function and then solving the potential inside the boundary with a fast Poisson solver. Since the solution of Poisson's equation is unique, our solution is exactly the same as the one obtained by simply using Green's function. The method allows us to select a much smaller region of mesh and therefore increase the resolution of the solver. The better resolution makes more accurate the calculation of the dynamics in the core of the beams. The luminosity simulated with this method agrees quantitatively with the measurement for the PEP-II B Factory ring in the linear and nonlinear beam current regimes, demonstrating its predictive capability in detail.

Beam-Beam Performance Of The SLAC B-Factory

“…Such an approach is self-consistent, because the electromagnetic field is computed by solving the Poisson equation with the charge distributions being updated as the beams collide. Recently, it was found that the extent of the x-y grid can be much reduced if an inhomogeneous potential is assigned on the boundary [5]. A smaller spatial extent allows for a denser mesh, thereby increasing the resolution of the Poisson solver.…”

Study of Beam-Beam Effects at PEP-II

“…Increased CPU power now allows full strong-strong simulations [5,7,8,9,10], where each bunch is modeled as a set of macroparticles. Each macroparticle is propagated through one turn of a storage ring using a linear matrix with damping and quantum excitations to the machine lattice.…”

“…At the interaction point (IP), the beam-beam effect is modeled by projecting the macroparticle distribution onto a transverse mesh, solving the Poisson equation on the mesh and evaluating the force that acts on the opposite beam. The method deployed here [8] uses a reduced fine mesh that covers only the central part of the beam pipe, large enough to contain the whole beam. The uniqueness of the Poisson solution is enforced by an appropriate choice of conditions at the mesh boundaries.…”

“…We first compute the luminosity and beam sizes by solving the Poisson equation numerically on a reduced mesh [8]. We then study the dependence of the specific luminosity on tunes, bunch currents, horizontal IP crossing angle and transverse beam separation, and compare the predicted values with those experimentally measured.…”

Terascale Beam-Beam Simulations for Tevatron, RHIC and LHC

“…A number of methods have been used to solve the Poisson equation. A five-point finite difference method with Fourier analysis and cyclic reduction (FACR) has been used by Krishnagopal [4] and Cai et al [5]. This method solves the Poisson equation efficiently with finite domain boundary conditions.…”