IJPAM: Volume 34, No. 1 (2007)

A MULTISYMPLECTIC GEOMETRY AND
A MULTISYMPLECTIC SCHEME FOR
MAXWELL'S EQUATIONS

Hongling Su$^1$, Mengzhao Qin$^2$, Rudolf Scherer$^3$
$^1$Department of Mathematics
Renmin University of China
Beijing, 100872, P.R. CHINA
e-mail: misssu007@yahoo.com.cn
$^2$State Key Laboratory of Scientific and Engineering Computing
Institute of Computational Mathematics and
Scientific/Engineering Computing
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
P.O. Box 2719, Beijing, 100080, P.R. CHINA
e-mail: qmz@lsec.cc.ac.cn
$^3$Institute for Practical Mathematics
University of Karlsruhe
Englerstrasse 2, Karlsruhe, 76128, GERMANY
e-mail: scherer@math.uni-karlsruhe.de


Abstract.In this paper the self-adjointness of Maxwell's equations with variable coefficients $\varepsilon$ and $\mu$ is discussed. For these equations three different Lagrangian forms are presented. Using Legendre's transformation, a multisymplectic Bridges' form is obtained. Based on the multisymplectic structure, the multisymplectic conservation law of the system is derived. A nine-point Preissman multisymplectic scheme, preserving the multisymplectic conservation law, is deduced for Maxwell's equations in an inhomogeneous, isotropic and lossless medium. A numerical example illustrates the results.

Received: August 25, 2006

AMS Subject Classification: 65P10, 70G55, 78A30

Key Words and Phrases: multisymplectic schemes, multisymplectic conservation laws, Maxwell's equations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 34
Issue: 1