Section 5

Applications in the Integral Forms Discurse of Mathematical Physics Equations

Chapter 5

Abstract

Electro-magnetic Simulation

Maxwell's Equations

reliable than those of the original FDTD method.

computational method

magnetic and electric flux densities

1. Introduction

63

Naofumi Kitsunezaki

Based on the Integral Form of

Algorithms for a computational method of electromagnetics based on the integral form of Maxwell's equations are introduced. The algorithms are supported by the lowest- and next-to-the-lowest-order approximations of integrals over a cell surface and edge of the equations. The method supported by the lowest-order approximation of the integrals coincides with the original finite-difference timedomain (FDTD) method, a well-known computational method of electromagnetics based on the differential form of Maxwell's equations. The method supported by the next-to-the-lowest-order approximation can be considered a correction to the FDTD method. Numerical results of an electromagnetic wave propagating in a two-dimensional slab waveguide using the original and the corrected FDTD methods are also shown to compare them with an analytical result. In addition, the results of the corrected FDTD method are also shown to be more accurate and

Keywords: Maxwell's equations, integral form, finite-difference time-domain method, the lowest-order approximation, next-to-the-lowest-order approximation,

Maxwell's equations are considered the fundamental equations of an electromagnetic field. They consist of laws of Faraday, Ampére-Maxwell, and Gauss for

where E and H are electric and magnetic fields, respectively, D and B are electric and magnetic flux densities, respectively, i is current density, ρ is charge density, <sup>∂</sup><sup>t</sup> <sup>f</sup> is the time derivative of field <sup>f</sup>, <sup>∇</sup> � <sup>A</sup> is the rotation of vector field <sup>A</sup>, <sup>∇</sup> � <sup>A</sup> is the divergence of vector A, and ρ is the electric charge density. Taking the

<sup>∂</sup>t<sup>B</sup> ¼ �<sup>∇</sup> � <sup>E</sup>, (1) <sup>∂</sup>t<sup>D</sup> <sup>¼</sup> <sup>∇</sup> � <sup>H</sup> � <sup>i</sup>, (2) ∇ � B ¼ 0, (3) ∇ � D ¼ ρ, (4)
