Upon discretization into a grid, (using various centralized difference, Crank–Nicolson method, FFT-BPM etc.) and field values rearranged in a causal fashion, the field evolution is computed through iteratio, along the propagation direction.
When discretization is commutative with dualization, then, under appropriate conditions, Pontryagin's minimum principle emerges as a consequence of the convergence of the discretization.
Mortar methods, discretization methods for partial differential equations