Symplectic Discretization of Hamiltonian Integral Equations with Applications to Nonlinear Wave Dynamics and Relativistic Orbits - Cheng, Pengcheng (Journal of nonlinear science, 2025)
We introduce a novel class of symplectic numerical schemes for solving nonlinear Volterra-type integral equations that arise from Hamiltonian systems subject to non-local interactions. The continuous integral equation is characterized by a Hamiltonian function H ( y ) and a kernel K ( t , s , y ) that satisfies a specific symplectic symmetry condition, K ( t , s , y ) = J ∂ S ( t , s , y ) ∂ y , ensuring the preservation of the phase space symplectic structure. Building upon this continuous framework, we develop high-order implicit symplectic integrators by discretizing the integral equation using an s -stage Gauss–Legendre collocation method. A rigorous proof demonstrates that the proposed numerical schemes are symplectic, i.e., they preserve the discrete symplectic form d y n + 1 ∧ J d y n + 1 = d y n ∧ J d y n . Furthermore, we establish the superconvergence property of these schemes, showing a global error of O ( h 2 s ) . For practical implementation, the resulting nonlinear algebraic equations at each time step are efficiently solved using a Newton–Krylov iterative method, and an adaptive time-stepping strategy based on embedded symplectic pairs is employed to enhance computational efficiency while maintaining numerical accuracy and symplectic fidelity. The superior long-term performance and structural preservation capabilities of the proposed methods are demonstrated through numerical experiments on challenging problems, including the soliton dynamics of the nonlinear Schrödinger equation and a model of relativistic orbital mechanics inspired by the Einstein–Infeld–Hoffmann equations. Comparisons with standard non-symplectic methods highlight the significant advantages of our symplectic approach in terms of energy conservation and phase space trajectory accuracy over extended simulation times.