Application of high order symplectic integration methods with forward integration steps in beam dynamics

The Hamiltonian describing particle motion in an accelerator belongs to a large class of systems, the members of which can be integrated with a new set of high order symplectic integrators. One benefit of these integrators is their strong numerical stability, which results from the inclusion of only forward integration steps, independent of the order of accuracy. Using these integrators, the transfer map of any multipolar accelerator magnet is derived and presented here. From these maps, the Hamiltonian flow in different lattices is simulated and benchmarked against other well established integration schemes in the accelerator community. By comparing quantities such as the linear phase advance and action invariant, the chromaticity, as well as the working point and the tune spread with amplitude, the superiority of the novel symplectic integrators is demonstrated with respect to accuracy and integration cost.

Figure 1

(a) The order of accuracy $\zeta$ for each element $M_{i,j}$ of the quadrupole transfer matrix is plotted for various integrators where the numbers in the labels inside the parentheses ($n_k$,$n_{\mathrm{tot}}$) refer to the amount of kicks $n_k$ and the total maps needed to construct the integrator $n_{\mathrm{tot}}$. The order of accuracy of $\mu$, in a $FODO$ cell that include dipoles, for different $K_Q$ and $L_Q$ with the use of (b) the $CSAB{A}_2$ and (c) the $TEAPO{T}_5$ symplectic integrators. The area below the white dashed curve guarantees stable motion through the $FODO$ cell.

Figure 2

The order of accuracy of $J$ in a $FODO$ cell, for different $K_Q$ and $L_Q$ with the use of (a) the $CSAB{A}_2$ and (b) the $TEAPO{T}_3$ symplectic integrators. The area below the white dashed curve guarantees stable motion through the $FODO$ cell.

Figure 3

The difference in order of accuracy of ${\xi }_x$ between (a) $CSAB{A}_2$ & $YF{R}_3$, (b) $CSAB{A}_2$ & $TEAPO{T}_3$, and (c) $CSAB{A}_4$ & $TEAPO{T}_5$. The difference in order of accuracy of ${\xi }_y$ between (d) $CSAB{A}_2$ & $YF{R}_3$, (e) $CSAB{A}_2$ & $TEAPO{T}_3$, and (f) $CSAB{A}_4$ & $TEAPO{T}_5$. The combinations of $(K_Q,L_Q)$ in the white area guarantee unstable motion and the black dots are configurations where the $CSAB{A}_{\nu }$ is accurate up to the 16th decimal digit of ${\xi }_x$ or ${\xi }_y$.

Figure 4

The difference in order of accuracy of $\frac{\partial Q_x}{\partial J_x}$ between (a) $CSAB{A}_2$ & $YF{R}_3$, (b) $CSAB{A}_2$ & $TEAPO{T}_3$, and (c) $CSAB{A}_4$ & $TEAPO{T}_5$. The difference in order of accuracy of $\frac{\partial Q_y}{\partial J_y}$ between (d) $CSAB{A}_2$ & $YF{R}_3$, (e) $CSAB{A}_2$ & $TEAPO{T}_3$, and (f) $CSAB{A}_4$ & $TEAPO{T}_5$. The combinations of $(K_Q,L_Q)$ in the white area guarantee unstable motion.

Figure 5

The variation of the $\mathcal{O}(\frac{\partial Q_j}{\partial J_j})$ for differed lattice configurations using (a) the $CSAB{A}_4$ and (b) the $TEAPO{T}_5$ symplectic integrators.

Figure 6

Tune diffusion resulted from the use of the $CSAB{A}_2$ symplectic integrator in (a) the coordinate space and (b) in the tune space. Tune diffusion resulted from the use of the $TEAPO{T}_3$ symplectic integrator in (c) the coordinate space and (d) in the tune space.

Figure 7

The difference in order of accuracy of $Q_x$ between (a) $CSAB{A}_2$ & $TEAPO{T}_3$ and (c) $CSAB{A}_2$ & $TEAPO{T}_5$. The difference in order of accuracy of $Q_y$ between (b) $CSAB{A}_2$ & $TEAPO{T}_3$ and (d) $CSAB{A}_2$ & $TEAPO{T}_5$.