High-Fidelity Simulations of Long-Term Beam-Beam Dynamics on GPUs

Futuremachines such as the Electron Ion Collider (MEIC), linac-ring machines (eRHIC) or LHeC are particularly sensitive to beam-beam effects. This is the limiting factor for long-term stability and high luminosity reach. The complexity of the non-linear dynamics makes it challenging to perform such simulations typically requiring millions of turns. Until recently, most of the methods have involved using linear approximations and/or tracking for a limited number of turns. We have developed a framework which exploits a massively parallel Graphical Processing Units (GPU) architecture to allow for tracking millions of turns in a sympletic way up to an arbitrary order. The code is called GHOST for GPU-accelerated High-Order Symplectic Tracking. Our approach relies on a matrix-based arbitrary-order symplectic particle tracking for beam transport and the Bassetti-Erskine approximation for the beam-beam interaction. INTRODUCTION AND BACKGROUND The proper magnetic optics design and performance of a storage ring or a collider—such as the LHC, RHIC, LHeC, and electron-ion colliders—crucially depends on its its longterm dynamics. Approaches which approximate the longterm dynamical stability based on relatively short-term simulations do not provide the necessary level of confidence. Ultimately, to simulate accurately the beam dynamics in a storage ring or a collider, it is necessary to track the beam particles for millions to billions of turns—comparable to the beam lifetime. However, until the recent advent of the GPU technology, such long-term simulations have been prohibitively expensive computationally. Long-term simulations require the tracking to be symplectic—invariants of motion must be explicitly preserved. A constant linear transfer map can be made trivially symplectic by ensuring that it satisfies the symplecticity criterion. Indeed, many “kick-drift” codes take advantage of this fact to perform a symplectic step-by-step integration of the particle’s equations of motion through a ring represented by a piecewise constant Hamiltonian. However, this approach is not suitable for long-term tracking due to the inherently large number of steps required for each particle turn around the ring. In order to attain the required efficiency, our new Gpu-accelerated Higher-Order Symplectic Tracking (GHOST) code uses a truncated single-turn non-linear Taylor map to track a particle while explicitly enforcing the ∗ bterzic@odu.edu symplecticity by solving a set of associated implicit, nonlinear set of equations. The beam collisions are described by the Poisson equation which can be solved by a number of methods at a high computational cost. To reduce the computational load, a number of approximations have been proposed. BEAMBEAM3D [1] uses a shifted integrated 2D Green’s function method to solve the equation on a grid. The 2D approximation is made possible by dividing the beams in thin slices. Another approximation can be to assume a gaussian beam distribution which leads to a one-dimensional integration [2]. Finally Bassetti-Erskine (BE) [3] approach introduces one more level of approximation by assuming that the beams have vanishing length and a Gaussian transverse distribution. This reduces the Poisson equation to a single evaluation of a complex error function, which is computationally efficient. In GHOST, we use the BE formalism for beam interaction, generalized to an arbitrary geometry, which may also include upright and round beams (as opposed to flat beams originally derived in BE). GHOST: ALGORITM DESCRIPTION In GHOST, the beam bunches are represented by an appropriate Gaussian distribution of point particles, while the effect of the collision is computed using the generalized BE approximation. The new code is SDDS-compliant [4], which can be readily post-processed with the powerful SDDS tools.