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 \cite{BCFLMM13} 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.

19 pages, 4 figures

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• Chaotic dynamics of graphene and graphene nanoribbons
• Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization
• Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
• Wave-packet spreading in disordered soft architected structures
• Frequency map analysis of spatiotemporal chaos in the nonlinear disordered Klein-Gordon lattice
• Energy spreading, equipartition and chaos in lattices with non-central forces
• Chaotic behaviour of disordered nonlinear lattices
• Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions