A second-order synchronized leapfrog advance scheme for coupled second-order ordinary differential equations

Summary form only given. A second-order symplectic Boris-like synchronized leapfrog scheme for advancing coupled second-order ODEs is developed. This scheme preserves second-order accuracy in advancing equations where acceleration (or its analogue) depends on velocity by inverting implicit equations in the synchronization step. The method is implemented in the WARP code to model the evolution of the first- and second- order moments, transverse to the direction of propagation, of charged particle beams subjected to periodic, skew-coupled electric and magnetic focusing.