[Paper Review] Multisymplectic Geometry Method for Maxwell's Equations and Multisymplectic Scheme

Hongling Su, Mengzhao Qin|ArXiv.org|Feb 24, 2003

Numerical methods for differential equations5 references3 citations

TL;DR

This paper develops a multisymplectic geometric framework for Maxwell’s equations in inhomogeneous, isotropic, lossless media by deriving a multisymplectic Hamiltonian form via Legendre transformation of a Lagrangian. It introduces a nine-point Preissman scheme that preserves the multisymplectic conservation law, demonstrating long-term stability and accuracy in numerical simulations of electromagnetic wave propagation.

ABSTRACT

In this paper we discussed the self-adjointness of the Maxwell's equations with variable coefficients $ε$ and $μ$. Three different Lagrangian are attained. By the Legendre transformation, a multisymplectic Bridge's (Hamilton) form is obtained. Based on the multisymplectic structure, the multisymplectic conservation law of the system is derived and a nine-point Preissman multisymplectic scheme which preserve the multisymplectic conservation law is given for the Maxwell's equations in an inhomogeneous, isotropic and lossless medium. At last a numerical example is illustrated.

Motivation & Objective

• To establish a self-adjoint variational formulation of Maxwell’s equations with variable ε and μ coefficients.
• To derive a multisymplectic Hamiltonian form using Legendre transformation from a Lagrangian density.
• To construct a structure-preserving numerical scheme that conserves the multisymplectic conservation law.
• To validate the scheme through numerical simulation of wave propagation with exact solutions.

Proposed method

• Derives three different Lagrangian densities for Maxwell’s equations in variable-coefficient media using inverse variational problem techniques.
• Applies the Legendre transformation to convert the Lagrangian system into a multisymplectic Hamiltonian form, revealing the underlying geometric structure.
• Derives the multisymplectic conservation law from the geometric structure of the Hamiltonian system.
• Constructs a nine-point Preissman multisymplectic scheme that preserves the multisymplectic conservation law by discretizing the Hamiltonian form directly.
• Uses auxiliary variables (U, V, P, Q) in the intermediate stage to derive the final nine-point scheme, which is then expressed in matrix form for numerical implementation.
• Employs a staggered leapfrog-type discretization in both space and time to maintain symmetry and conservation properties.

Experimental results

Research questions

• RQ1 Under what conditions is Maxwell’s equation in variable-coefficient media self-adjoint, enabling a variational formulation?
• RQ2 How can a multisymplectic Hamiltonian structure be derived from the Lagrangian of Maxwell’s equations?
• RQ3 What is the form of the multisymplectic conservation law in the context of Maxwell’s equations with spatially varying ε and μ?
• RQ4 Can a numerical scheme be constructed that preserves the multisymplectic structure while accurately simulating electromagnetic wave propagation?
• RQ5 How does the performance of the nine-point multisymplectic scheme compare to exact solutions in long-time simulations?

Key findings

• The paper establishes necessary and sufficient conditions for self-adjointness of Maxwell’s equations in a 1st-order form with variable ε and μ, enabling variational formulation.
• Three distinct Lagrangians are derived for Maxwell’s equations, depending on the choice of field representation.
• A complex multisymplectic Hamiltonian form is obtained via Legendre transformation, revealing the system’s intrinsic geometric structure.
• The nine-point Preissman scheme is shown to exactly preserve the multisymplectic conservation law, ensuring long-term stability.
• Numerical results for ε = μ = 1 and J = 0 show excellent agreement with the exact solution, with the numerical solution maintaining wave-like behavior and minimal error growth over 1000 time steps.
• The absolute error at t = 1000 exhibits a regular, oscillatory pattern closely matching the shape of the exact solution, indicating high accuracy and structural fidelity.