Nonstandard FDTD Theory

Beginner Course

1. Finite Difference Model
2. 1D Wave Equation
3. 1D Stability
4. 1D Boundary Condition
5. Dimensionless Form
6. 1D Example

Intermediate Course

1. 2D Wave Equation
2. 2D Maxwell's Equations
3. Scattered Field
4. 2D Stability
5. 2D Boundary Condition
6. Scatterer Representation
7. Intensity
8. 2D Example

Advanced Course

1. 3D Wave Equation
2. 3D Maxwell's Equations
3. 3D Stability
4. Conductive Media
5. Perfectly Matched Layer

2. 3D Maxwell's Equations

Maxwell's equations are given by

Let us find a S- and NS-FDTD algorithms for this wave equation.

Standard Yee Algorithm

Fig. 1. Yee cell in three dimensions. h is the grid spacing. E_x, E_y, E_z are electric field components. H_x, H_y, H_z are magnetic field components. Each component alternately lies on the numerical grid.

In three dimensions, as shown in Fig. 1, the Yee cell is defined by

Using the S-FDTD algorithm in the Yee cell, the Maxwell's equations are discretized as

where d_S = (d_x, d_y, d_z). Expanding the temporal difference operators, we obtain a "3D S-Yee algorithm,"

Nonstandard Yee Algorithm

In the similar way to two dimensions, we define

which satisfied d_NS² = d_S⋅d_NS = d_NS⋅d_S. Using (5), we define

where Z = (μ/ε)^1/2. Since (6) is equivalent to the NS-FDTD algorithm for the 3D wave equation, it is a high accuracy form optimized to a monochromatic wave. Expanding the temporal difference operators and using

we obtain a "3D NS-Yee algorithm,"

Copyright (C) 2011 Naoki Okada, All Rights Reserved.