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Beam dynamics topics
Published in Xiaobiao Huang, Beam-based Correction and Optimization for Accelerators, 2019
The general idea of explicit symplectic integration is to split the Hamiltonian into two integrable parts, such as a drift space and a lumped kick. In each integration step, a number of drifts and kicks are alternately applied to the particles. Since transporting through drifts and thin-lens kicks are both symplectic, the total transport is automatically symplectic. If the lengths of the drifts, the strengths of the kicks, and the order of application are properly chosen (independent of the actual Hamiltonian), the integration will be accurate as well as symplectic. A simple case is the second order symplectic integrator illustrated in Figure 2.7. Each integration step consists of a drift, a kick, and another drift. The lengths of the drifts are equal to one half of the step length and the kick corresponds to the integrated magnetic field over the step length. The symmetric configuration eliminates the first order errors such that the leading error terms are O(L2). Slicing the element into many integration steps will increase the accuracy of the solution.
The FDTD Method: Essences, Evolutions, and Applications to Nano-Optics and Quantum Physics
Published in Sarhan M. Musa, Computational Nanotechnology Using Finite Difference Time Domain, 2017
Xiaoyan Y.Z. Xiong, Wei E.I. Sha
The high-order symplectic integration scheme has desired numerical precision and high numerical stability but needs multiple stages in every time step. Compared with the R-K method, it has the energy-preserving property and saves memory.
Numerical Methods: Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Symplectic integration An integration method is symplectic if the state of the (Hamiltonian) system following an integration step could have been reached from that before the step by some canonical transformation. The most straightforward way to test if a method is symplectic is to verify the Poisson bracket relations between the before and after states. Given a method that determines u(x), where u and x are both s-dimensional, let J be the s×s Jacobian matrix that leads from “before” to “after”: J=∂(un,xn)∂(un−1,xn−1). Now define the matrix K=0sIs−Is0s. If JTKJ=K, then the method is symplectic. See page 560.
Practical perspectives on symplectic accelerated optimization
Published in Optimization Methods and Software, 2023
Valentin Duruisseaux, Melvin Leok
Symplectic integrators form a class of geometric numerical integrators of interest since, when applied to Hamiltonian systems, they yield discrete approximations of the flow that preserve the symplectic 2-form. The preservation of the symplectic 2-form results in the preservation of many qualitative aspects of the underlying dynamical system. In particular, the numerical solution of a Hamiltonian system obtained using a constant time-step symplectic integrator is exponentially-near to the exact solution of a nearby Hamiltonian system for an exponentially-long time [14,43]. It explains why symplectic integrators exhibit good energy conservation with essentially no accumulation of errors in time, when applied to Hamiltonian systems, and why symplectic methods are best suited to integrate Hamiltonian systems. We refer the reader to [45] for a brief recent overview of geometric numerical integration, and to [17,43,53] for a more comprehensive presentation of structure-preserving integration techniques.