Symplectic Error of Implicit Symplectic Integrators: A Qualitative Structural Analysis
Mat\v{e}j Gajdo\v{s}, Ond\v{r}ej Brichta, V\'aclav Ku\v{c}era

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 of fixed-point iterations (FPI). While SE is exactly symplectic under exact solves, a finite 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 . We prove that the perturbed matrix is skew-symmetric, that one diagonal block vanishes identically (depending on the SE variant), and that the remaining blocks are perturbations of their counterparts in , with time step . A…
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