High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schrödinger equation

• Hamiltonian systems that split in three integrable parts are considered.
• For such systems, several high order three part symplectic integrators (SIs) based on composition methods are reviewed.
• The different SIs are benchmarked by a practical Hamiltonian. Non-symplectic integrators are also compared.
• This Hamiltonian dictates asymptotic wavepacket spreading in the disordered discrete nonlinear Schrödinger equation.
• Three part split SIs are shown more efficient than other symplectic and non-symplectic methods, in both accuracy and CPU time.

While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we present several high order symplectic integrators for Hamiltonian systems that can be split in exactly three integrable parts. We apply these techniques, as a practical case, for the integration of the disordered, discrete nonlinear Schrödinger equation (DDNLS) and compare their efficiencies. Three part split algorithms provide effective means to numerically study the asymptotic behavior of wave packet spreading in the DDNLS – a hotly debated subject in current scientific literature.