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

4. 2D Stability

The numerical stability of the NS-Yee algorithm for the Maxwell's equations is the same as the NS-FDTD algorithm for the wave equation, because they are equivalence in homogeneous media. As shown in the 1D stability, the numerical stability for the wave equation is given by

where D is defined below.

S-FDTD Stability

In the 2D S-FDTD algorithm, D is defined by

For the monochromatic wave, ψ(x,y,t) = e^{i(k_xx+k_yy±ωt)}, we obtain

To compute the maximum value of D², we solve

For cos(k_xh/2) = cos(k_yh/2) = 0, D² is maximized as

Thus, the stability of the S-FDTD algorithm becomes

where we use h = Δt = 1.

NS-FDTD Stability

In the 2D NS-FDTD algorithm D² is given by

where

To obtain max(D²), we solve

Substituting (9) into (8), we obtain

Thus, the stability of the NS-FDTD algorithm is

where we use k ∼ 0 and h = Δt = 1. In two dimensions, comparing with the S-FDTD stability, the value of v can be increased by about 10%.

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