Numerical methods in high-intensity linacs

About

Numerical methods were historically developed to tackle equations that resist analytical solutions—first in celestial mechanics, where the complexity of planetary orbits defied closed-form answers, and today in modern physics applications such as particle accelerators. In high-intensity linacs, many collective effects, foremost among them space charge, give rise to nonlinear dynamics that are analytically intractable and must be addressed through numerical integration. This lecture provides an overview of the key numerical techniques used to model beam dynamics in particle accelerators. We will review classical methods such as Runge–Kutta and Störmer–Verlet, examining their strengths and limitations in the context of multi-particle tracking and self-consistent field evolution. The focus will then shift to geometric integration methods, which are specifically designed to preserve fundamental physical invariants of Hamiltonian systems, such as phase-space volume and symplectic structure. Among these, we will highlight Lie operator splitting techniques and the Yoshida symplectic integrator, discussing their derivation, implementation, and practical advantages in accelerator simulations.

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