Symplectic Error of Implicit Symplectic Integrators: A Qualitative Structural Analysis

TL;DR

This paper analyzes how inexact nonlinear solvers affect the symplectic properties of implicit symplectic integrators, providing a detailed qualitative characterization of the resulting pseudo-symplecticity and energy errors.

Contribution

It offers a novel block-wise analysis of pseudo-symplecticity induced by fixed-point iterations in implicit schemes, extending understanding of volume preservation and energy errors.

Findings

Perturbed matrix $ ilde{J}$ remains skew-symmetric with specific block structures.

Bounds on perturbations are sharp, confirmed by quadratic Hamiltonian example.

Numerical experiments validate theoretical bounds and reveal gaps to exact symplecticity.

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.