Nonstandard finite difference methods for Hamiltonian systems

Numerical methods for Hamiltonian systems must preserve conservation laws to maintain solution quality over long integration times. Even with low convergence order, such structure-preserving schemes often outperform general-purpose high-order methods by avoiding both artificial dissipation and unphysical energy drift. There exist two principal paradigms, namely: energy-preserving methods, which exactly conserve the Hamiltonian, and symplectic methods, which preserve the geometric structure of phase space. It has been proven that no numerical integrator can simultaneously satisfy both properties for non-integrable systems. Energy-preserving methods are typically implicit and incur substantially higher computational costs per time step. Symplectic integrators conserve the modified Hamiltonian, thus exhibiting no systematic energy drift over long-time simulations. Additionally, numerical integrators generate phase-lag or phase-lead, which accumulates over time and inevitably results in energy drift when symplecticity is not maintained.
In this work, to improve the numerical phase portrait of Hamiltonian systems, two modifications of the Verlet method are proposed using the Nonstandard Finite Difference (NSFD) methodology. The researchers refer to these modifications as the elliptic and hyperbolic Verlet methods. For completeness, the standard Verlet method is termed parabolic. The elliptic and hyperbolic methods are distinguished by either contracting or prolonging the step-size depending on a tunable parameter. By optimally selecting this parameter, one can reduce the phase-lag or phase-lead and, consequently, improve energy conservation relative to the standard Verlet method. The choice among the three integrators—elliptic, hyperbolic, and parabolic—depends on the topology of the potential energy at the initial position. The proposed methods are extended to multi-dimensional Hamiltonian systems, providing a novel feature: individual step-size control along each phase variable. This feature is particularly advantageous for problems where fast and slow dynamics occur simultaneously.