Structural Inconsistency and Stability Classification of Multi-symplectic Diamond Schemes

TL;DR

This paper develops a systematic framework to analyze the stability and structural consistency of multi-symplectic diamond schemes for Hamiltonian PDEs, revealing their limitations and stability conditions.

Contribution

It introduces a comprehensive stability analysis method combining structural inconsistency detection, graph-based error propagation, and eigenvalue analysis for multi-symplectic schemes.

Findings

Diamond schemes are incompatible with some benchmark equations like KdV.

Stability regimes exist for certain equations such as nonlinear Dirac.

The framework predicts when diamond schemes are structurally inconsistent or unstable.

Abstract

Multi-symplectic diamond schemes proposed by McLachlan and Wilkins (2015) provide a framework for the numerical integration of Hamiltonian partial differential equations, combining local implicitness with high-order accuracy and discrete multi-symplectic conservation laws. Despite these advantages, their behavior beyond a limited class of model equations remains poorly understood, and numerical difficulties may arise depending on the underlying multi-symplectic formulation. In this paper, we present a systematic stability analysis framework for diamond schemes applied to general multi-symplectic PDEs. The approach consists of three stages. First, we identify structural inconsistency of the local diamond update using Dulmage--Mendelsohn decomposition, revealing cases in which the scheme is intrinsically unsolvable. Second, we introduce a graph-based error-propagation analysis that yields a necessary stability condition by detecting negative cycles in a weighted directed graph. Third, for equations that pass the preliminary tests, we derive eigenvalue-based timestep restrictions providing sufficient conditions for stability. The analysis leads to a comprehensive classification of multi-symplectic PDEs according to whether diamond schemes are structurally inconsistent, unconditionally unstable, or conditionally stable. In particular, we show that benchmark equations such as the Korteweg--de Vries equation are intrinsically incompatible with the diamond update, while systems including the nonlinear Dirac and ``good'' Boussinesq equations admit stability regimes under mild timestep scaling. Extensive numerical experiments confirm the theoretical predictions and demonstrate the practical implications of the proposed framework. Our results clarify fundamental limitations of diamond schemes and provide practical guidelines for their reliable application to new PDE models.