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, \(K(t, s, y) = J \frac{\partial S(t, s, y)}{\partial y}\) , 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 \(d y_{n+1} \wedge J d y_{n+1} = d y_n \wedge J d y_n\) . Furthermore, we establish the superconvergence property of these schemes, showing a global error of \(O(h^{2s})\) . 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.
