Solving Maxwell's Equations with Mimetic Methods

Abstract

We present a mimetic finite-difference approach for solving Maxwell's equations in one and two spatial dimensions. After introducing the governing equations and the classical Finite-Difference Time-Domain (FDTD) method, we describe mimetic operators that satisfy a discrete analogue of the extended Gauss divergence theorem and show how they lead to a compact, physically consistent formulation for computational electromagnetics. Two numerical examples are presented: a one-dimensional sinusoidal wave interacting with a lossy dielectric slab, and a two-dimensional Gaussian pulse with Uniaxial Perfectly Matched Layer (UPML) absorbing boundary conditions. All implementations use the Mimetic Operators Library Enhanced (MOLE).

Figures (7)

• Figure 1: One-dimensional Yee staggered grid. Electric field values (dots) are stored at integer nodes; magnetic field values (crosses) at half-integer nodes.
• Figure 2: One-dimensional mimetic staggered grid with $m=4$ cells. Scalar values (dots) reside at cell centers and boundary nodes ($m+2$ locations). Vector values (ticks) reside at cell edges ($m+1$ locations).
• Figure 3: Two-dimensional staggered grid ($m=4$, $n=3$). Scalar values (dots) at cell centers and boundaries; vector components (ticks perpendicular to cell sides) at cell edges.
• Figure 4: Electric and magnetic fields after 500 steps for a 700 MHz sinusoidal wave interacting with a lossy dielectric slab ($\varepsilon_r = 4$, $\sigma = 0.04$ S/m).
• Figure 5: Initial electric field ($E_z$) for a 2D Gaussian pulse with UPML absorbing boundaries ($d_{\text{PML}} = 30$ cells, $\sigma_{\max} = 100$, $p = 4$).