The time-dependent wave equation can be solved numerically by an iterative time-marching algorithm called the Finite Difference Time Domain (FDTD) algorithm. We introduce FDTD for the wave equation. There is a subset of light diffraction problems where the Wave equation can be used instead of Maxwell’s equations. Identifying these cases is important since FDTD with the Wave Equation can greatly reduce the computation time compared to FDTD with Maxwell’s Equations.
To derive an FDTD algorithm, we need to model the differential equation by replacing the derivatives with finite difference (FD) expressions. We review the different kinds of FD expressions and introduce a FD model of the wave equation, and then the FDTD algorithm. We analyze its accuracy and numerical stability, and find that what is called the nonstandard (NS) FD model can greatly reduce the error. We implement FDTD in Python programs, and solve some example problems.
Key concepts covered include:
- Finite differences and finite difference models of differential equations
- Finite difference model of the wave equation
- Model set up in one dimension and choosing the parameters
- Accuracy, numerical stability and other constraints on algorithms parameters
- Computational boundaries: periodic, reflecting, absorbing
- Simulating an infinite space, a periodic structure, and a cavity
- Object representation on the grid for FDTD simulations
- Applicability of the Wave Equation in solving diffraction problems
- Coding in Python/Matlab and simulation training with FDTD