Numerical Methods
There are both spatial domain methods, and frequency (spectral) domain methods are available for the numerical solution of the discretized master equation. Upon discretization into a grid, (using various centralized difference, CrankâNicolson method, FFT-BPM etc.) and field values rearranged in a causal fashion, the field evolution is computed through iteratio, along the propagation direction. The spatial domain method computes the field at the next step (in the propagation direction) by solving a linear equation, whereas the spectral domain methods use the powerful forward/inverse DFT algorithms. Spectral domain methods have the advantage of stability even in the presence of nonlinearity (from refractive index or medium properties), while spatial domain methods can possibly become numerically unstable.
Read more about this topic: Beam Propagation Method
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