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

3. 1D Stability

We wish v = ω/k is as large as possible to reduce computation time in the simulaion, but v is constrained by a numerical stability. Let us derive the stability for electromagnetic waves and find the maximum of v.

For a monochromatic wave, ψ(x,t) = e^i(kx±ωt), we obtain

Then, the S-FDTD and NS-FDTD algorithms are rewritten by

where

These algorithm stabilities are given by |λ_±| ≤ 1 (λ_± = eigenvalues of A), because Aⁿ is expanded into

where P consists of eigenvectors and λ_± is found by solving

where I is the identity matrix. We obtain

Since |λ_±| ≤ 1 derives |Z ≤ 1, we find

The stability is absolutely stabilized on the worst case, max(D²) = 4. thus, the stability of 1D S- and NS-FDTD algorithms become

The smaller Δx, not only the higher accuracy but the smaller v indicates increasing the computation time.

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