Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

Abstract

We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one-and two-dimensional disordered Klein-Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems' maximum Lyapunov exponent estimator. Among the tested integrators the fourth order ABA864 scheme [59] showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems' total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme s11ABA82 6 exhibited the best performance.

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

“…It is worth noting that two part split SIs of order six and higher often do now not produce reliable results for relative low energy accuracies like E r ≈ 10 −5 for the α-FPUT system (similar behaviors were reported in [56] for the DKG model). This happens because the required integration time step τ needed to keep the relative energy error at E r ≈ 10 −5 is typically large, resulting to an unstable behavior of the integrator i.e.…”

“…From the results of this figure we see that X 1 exhibits a tendency to decrease following a completely different decay from the X 1 ∝ t −1 power law observed for regular motion. This behavior was also observed for the 2D DKG model [56], as well as for the 1D DKG and DDNLS systems in [32,33] where a power law X 1 (t) ∝ t α L with α L ≈ −0.25 and α L ≈ −0.3 for, respectively, the weak and strong chaos dynamical regimes was established. Further investigations of the behavior of the finite mLE in 2D disordered systems are required in order to determine a potentially global behavior of X 1 , since here and in [56] only some isolated cases were discussed.…”

“…We note that we presented here a detailed comparison of the performance of several two and three part split SIs for the integration of the variational equations through the tangent map method and consequently for the computation of a chaos indicator (the mLE), generalizing, and completing in some sense, some sporadic previous investigation of the subject [56,58,59,60], which were only focused on two part split SIs. We hope that the clear description of the construction of several two and three part split SIs in Sec.…”

“…q 0 = q N , p 0 = p N , δq 0 = δq N , δp 0 = δp N , q N+1 = q 1 , p N+1 = p 1 , δq N+1 = δq 1 , δp N+1 = δp 1 . In the case of fixed boundary conditions we considered in our numerical simulations, some adjustments have to be done for the i = 1 and i = N equations, like the ones reported in the Appendix of [56] where the operators for the 1D and 2D DKG models were reported (with the exception of the corrector term). For completeness sake in Sec.…”

“…i, j = q i−1, j + q i, j−1 + q i, j+1 + q i+1, j q i, j = 0 δp i, j = δq i−1, j + δq i, j−1 + δq i, j+1 + δq i+1, j δq i, j = 0 , (A. 22) while the corresponding operators e L B 2 Z and e L C 2 Z are respectively e τL B 2 Z : [56] for the 2D DKG lattice should be performed.…”