Multisymplectic Geometry Method for Maxwell's Equations and Multisymplectic Scheme

TL;DR

This paper develops a multisymplectic geometric framework for Maxwell's equations with variable coefficients, deriving conservation laws and proposing a numerical scheme that preserves these laws for inhomogeneous media.

Contribution

It introduces a multisymplectic approach to Maxwell's equations with variable coefficients and constructs a structure-preserving numerical scheme.

Findings

Derived multisymplectic conservation law for Maxwell's equations.

Proposed a nine-point Preissman multisymplectic scheme.

Numerical example demonstrating scheme effectiveness.

Abstract

In this paper we discussed the self-adjointness of the Maxwell's equations with variable coefficients $\epsilon$ and $\mu$. 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.