Beam-RF Simulation via Partial Decomposition of the Maxwell Equations Part I: Mathematical Framework

Simulating the interaction of RF-waves with charged particle beams is difficult and computationally expensive, even on present high-end computers. There is a tradeoff between fidelity and complexity/expense of the two primary techniques, particle-in-cell and envelope methods. This paper presents an approach based on the partial decomposition of the Maxwell equations to achieve a fully second-order set of ordinary differential equations (ODEs) describing the interaction. It has the potential for self-consistent inclusion of ac and dc space-charge effects through a solution of evanescent ODEs, effects that are handled poorly, if at all, by existing envelope methods. The resulting technique provides flexibility in adjusting simulation fidelity versus computational costs. The developments presented in this paper establish a mathematical framework for this technique and test its core aspects by demonstrating accurate solution of the ODEs using analytic sources.