Numerical Dispersion in FDTD Method: Analysis and Code Implementation

The numerical dispersion problem in FDTD (Finite-Difference Time-Domain) method represents a classic challenge in electromagnetic field simulation. Due to spatial and temporal discretization in FDTD algorithms, the simulated electromagnetic wave propagation speed deviates from actual physical conditions. This frequency-dependent error phenomenon is termed numerical dispersion.

Typical FDTD simulations exhibit three key characteristics: Firstly, high-frequency components propagate slower than low-frequency components, which contrasts with physical dispersion in real media. Secondly, the degree of numerical dispersion directly correlates with grid size - when grid resolution reaches λ/20 or finer, errors become significantly reduced. Finally, different propagation directions produce anisotropic dispersion properties that can be analyzed using directional wave propagation tests in code implementation.

When validating this phenomenon in MATLAB, practitioners typically establish plane wave propagation within computational domains. The dispersion magnitude is quantified by comparing time differences of different frequency components arriving at detection points, often implemented through Fourier analysis of time-domain signals. Improvement strategies include implementing higher-order difference schemes (e.g., 4th-order spatial derivatives), non-uniform grid techniques using graded meshing functions, or sub-gridding methods with interface boundary conditions. Understanding numerical dispersion effects is crucial for proper simulation parameter configuration and result interpretation, particularly in broadband simulations and fine-structure modeling scenarios where dispersion errors can accumulate and distort simulation accuracy.