1D, 2D, and 3D Finite-Difference Time-Domain (FDTD) Methods: Implementation and Applications

The Finite-Difference Time-Domain (FDTD) method is a widely used numerical technique for simulating electromagnetic wave propagation by discretizing and solving Maxwell's equations. Based on spatial dimensions, it can be implemented in three variants: 1D, 2D, and 3D.

1D FDTD serves as the fundamental implementation, ideal for linear structures or rapid algorithm validation. It models electromagnetic field variations along a single direction (e.g., X-axis) with minimal computational requirements, clearly demonstrating field evolution through time stepping. Code implementation typically involves simple arrays for field storage and central-difference approximations for spatial derivatives.

2D FDTD extends to planar analysis with two polarization modes: TM (Transverse Magnetic) and TE (Transverse Electric). Utilizing Yee grid spatial staggering for electric and magnetic field components, it balances computational efficiency and accuracy for scenarios like waveguide analysis or planar antennas. Implementation requires 2D matrix operations and careful handling of field component interleaving.

3D FDTD provides the most comprehensive implementation using cubic Yee grids to fully simulate electromagnetic interactions in three-dimensional space. Despite significantly increased computational complexity, it accurately models complex structures like antenna arrays or bio-electromagnetic effects. The code structure employs 3D array manipulations and optimized memory management for large-scale simulations.

Sullivan's seminal work provides classic MATLAB implementation examples featuring three core components: grid initialization, time-loop field updates, and boundary condition handling (e.g., PML absorbing boundaries). The implementation emphasizes educational value through modular design, highlighting the core FDTD concept of leapfrog time-stepping between electric and magnetic fields. Key functions often include field update equations using curl operations and coefficient pre-calculation for performance optimization.

In practical applications, dimension selection depends on problem requirements: 1D for rapid principle verification, 2D for in-plane propagation analysis, and 3D for full-space simulation. MATLAB's matrix operation capabilities naturally suit FDTD's finite-difference calculations for spatial derivatives, though 3D implementations require careful memory management and potential parallelization strategies for large-scale problems.