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Structural Inconsistency and Stability Classification of Multi-symplectic Diamond Schemes

Kaito Sato, Shun Sato, Takayasu Matsuo

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.

Structural Inconsistency and Stability Classification of Multi-symplectic Diamond Schemes

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.

Paper Structure

This paper contains 35 sections, 5 theorems, 80 equations, 38 figures, 1 table.

Table of Contents

  1. Introduction
  2. A brief introduction of diamond schemes
  3. The simple diamond scheme
  4. Updating a single diamond
  5. Overall updating procedure
  6. The diamond scheme: high-order schemes using Runge--Kutta collocation methods
  7. Properties of diamond schemes
  8. Step 1: Detecting structural inconsistency of diamond schemes
  9. Motivating examples
  10. Equation--unknown bipartite graph and DM decomposition
  11. Step 2: Estimating the (in)stability
  12. Error propagation: Observation in the wave equation
  13. Error propagation: General cases
  14. Mixed derivative type Klein--Gordon equation
  15. Improved Boussinesq equation
  16. ...and 20 more sections

Key Result

Proposition 2

Consider a linear multi-symplectic PDE, and suppose the simple diamond scheme is structurally inconsistent on it. Then the coefficient matrix $(K/\Delta t-P/4)$ is singular.

Figures (38)

  • Figure 1: Flowchart of the stability analysis process for the diamond schemes
  • Figure 2: Diamond-shaped mesh
  • Figure 3: A single diamond
  • Figure 4: A transformed diamond for the (higher-order) diamond schemes using Runge--Kutta collocation methods
  • Figure 5: Dulmage--Mendelsohn decomposition of the bipartite graph for the advection equation: equations vs. unknowns $(\phi^{t},u^{t},w^{t})$.
  • ...and 33 more figures

Theorems & Definitions (13)

  • Definition 1: Structural inconsistency
  • Proposition 2: Structural inconsistency implies singular coefficient matrix
  • proof
  • Proposition 3
  • proof
  • Definition 4: Error-propagation graph (reduced version)
  • Theorem 5
  • Lemma 6
  • proof
  • Definition 7: Error-propagation graph (full version)
  • ...and 3 more