Symplectic Error of Implicit Symplectic Integrators: A Qualitative Structural Analysis

Abstract

We study how inexact nonlinear solvers lead to a loss of exact symplecticity in the Symplectic Euler (SE) and Stormer-Verlet (SV) schemes when applied to general nonseparable Hamiltonian systems. These schemes are implicit and require nonlinear solvers in practice. Here, we consider a fixed number $M$ of fixed-point iterations (FPI). While SE is exactly symplectic under exact solves, a finite $M$ gives only pseudo-symplecticity. Compared to previous results, we provide a more qualitative, block-wise characterization of the induced pseudo-symplecticity by analyzing the resulting perturbations to the matrix of symplectic structure $J$. We prove that the perturbed matrix $\tilde{J}$ is skew-symmetric, that one diagonal block vanishes identically (depending on the SE variant), and that the remaining blocks are $O(h^{M+1})$ perturbations of their counterparts in $J$, with time step $h$. A quadratic Hamiltonian example shows these bounds are sharp. Extending to compositions, we quantify how SV inherits distinct decay orders across different blocks of the symplectic defect. As a corollary, we show that the perturbation of volume preservation in phase space arises solely from the off-diagonal blocks of $\tilde{J}$, and we bound the induced energy error along trajectories. Numerical experiments on a tokamak magnetic-field Hamiltonian, where q-implicit SE is fully nonlinear (requiring FPI) but p-implicit SE is linearly implicit, confirm the sharpness of the theory and highlight the gap to the exactly symplectic counterpart.