[Paper Review] Symplectic Mimetic Finite Difference methods for Hamiltonian wave equations in 2D

L. Beirão da Veiga, L. Lopez+1 more|arXiv (Cornell University)|May 5, 2015

Numerical methods for differential equations3 citations

TL;DR

This paper presents a symplectic mimetic finite difference method for solving Hamiltonian wave equations in two spatial dimensions. By combining mimetic spatial discretization—preserving key structural properties of the continuous system—with symplectic time integration, the method conserves energy and momentum over long-time simulations, ensuring high accuracy and stability in Hamiltonian systems.

ABSTRACT

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.

Motivation & Objective

• To develop a numerical method that preserves the geometric structure of Hamiltonian wave equations in two spatial dimensions.
• To address the challenge of long-time numerical instability in wave equations by maintaining symplecticity and mimetic properties.
• To combine mimetic finite differences in space with symplectic time integrators for enhanced conservation and accuracy.
• To provide a theoretical analysis and numerical validation of the proposed method's stability and convergence.

Proposed method

• The spatial discretization uses mimetic finite difference (MFD) methods to preserve fundamental properties of the continuous Hamiltonian system, such as conservation laws and boundary conditions.
• The semi-discrete system resulting from MFD spatial approximation is advanced in time using a symplectic integrator to maintain the system's geometric structure.
• The method ensures that the discrete system inherits key properties like skew-symmetry and consistency with the continuous operator.
• The formulation is designed to maintain momentum and energy conservation over long-time simulations.
• A complete theoretical analysis is performed to verify stability, convergence, and structure preservation of the full scheme.

Experimental results

Research questions

• RQ1 Can mimetic finite differences combined with symplectic time integration preserve the Hamiltonian structure of 2D wave equations?
• RQ2 How well does the method conserve energy and momentum over long integration times?
• RQ3 What is the convergence behavior and stability of the proposed method in comparison to standard finite difference schemes?
• RQ4 Does the method maintain accuracy for problems with complex boundary conditions?

Key findings

• The proposed method successfully preserves the Hamiltonian structure of the 2D wave equation through both spatial and temporal discretization.
• Numerical simulations demonstrate long-term energy conservation, a hallmark of symplectic methods, even over extended time intervals.
• The mimetic spatial discretization ensures consistency with the continuous system's conservation laws and boundary behavior.
• Theoretical analysis confirms the method's stability and convergence under standard assumptions.