Assignment Problem:
This is a problem for writing an FDTD code and understanding various numerical aspects related to FDTD simulations. You can use any programming language and environment.
Problem Description: Consider a set of TEz (TEMz) 1D Maxwell's equations with the computational domain 0 < x < xmax. The fields are excited by specifying the electric field at x = 0 with two time dependence types:
where σ =1, t0 = 5σ are quantities normalized for the speed of light of c =1.
Assignment:
Choose Δx
Choose Δt = γ Δx/c, where
- γ = 1
- γ = 0.99
- γ = 0.5
Set up numerical equations of FDTD and write a numerical code to compute the electric field E and magnetic field H in space and time.
Show results for the electric field E for the time dependences (a) and (b) for the three values of γ each (total 6 figures) as a function of x for t = 20σ.
Explain the differences in the field for (a) and (b) and for different γ, e.g. are there any ripples in the field space and time dependence, are there any distortions, reasons for the behavior, ways to improve etc.
Now for the time dependence (b), choose a sequence of the values of Δx and show the error of computing the electric field at t = 20σ as a function of Δx. The error is defined as
where F(tn - xi/c) is the function from the time dependence (b), which is also an analytical solution of the 1D Maxwell's equations at time tn and space locations xi, and E(tn - xi/c) is the numerical solution via FDTD.