Finite Difference & FDTD

Prepare a .pdf file presenting your work in this assignment and upload it to Canvas.

1. The finite difference for Poisson Equation and Capacitance of Square Coaxial Cable

Objective: Reformulate the equations for Poisson Equation as a linear system in matrix form. Your task is to create the A matrix and g vector, and then solve the linear system using Phi = A\g Matlab command

Instructions:

• Check the slides, the Matlab code and Section 3.1.3 of the Textbook where we solved the problem using the Gauss-Seidel iteration that doesn't require matrix formulation
• Formulate the problem as matrix problem (it might be helpful to check the slides on finite differences)
  • What is the equivalent of symmetry BC in matrix form?
• Plot the Potential on the grid and compare it with the result of the last iteration from the code we used in class.
• Perform a convergence test and plot it
• Determine experimentally the order of truncation error

2. FDTD

Objective: implement in Matlab the discretization of the second-order (or curl-curl) formulation of Maxwell's equations in 1D for the propagation of two light waves from a pulse source. Compare the results obtained with the Yee Scheme. Determine experimentally the numerical dispersion relations.

Instructions:

• Download and run the code using the Yee scheme describing for the propagation of two light waves from a pulse source. A pulse is initialized in the central point of the simulation domain.  The pulse amplitude increases exponentially in time until t=t₀ and then decreases exponentially after that.
• Read Section 5.1 of the Textbook
• Remove the calculation of E and H in the code and implement the computation of the E using the curl-curl formulation (Section 5.1 of the Textbook)
• Plot E at different time steps for the two methods using
  1. a time step matching the Courant condition
  2. a time step allowing stable simulation but not perfectly matching the Courant condition
• Determine experimentally the dispersion relation using a 2D FFT (space and time) of the history of Ez. For doing that, you can

ExHist = [];
% Start loop
for t=1:nsteps
   % E field loop
   for k=2:ke-1
    Ez(k)=Ez(k)+cc*(Hy(k-1)-Hy(k));
   end

% Source
   Ez(ks)=exp(-.5*((t-t0)/spread)^2);
   % H field loop
   for k=1:ke-1
      Hy(k)=Hy(k)+cc*(Ez(k)-Ez(k+1));
   end
   ExHist = [ExHist; Ez];
end

spectrum = fft2(ExHist);
plot_spectrum = log(abs(spectrum));
pcolor(plot_spectrum(1:ke/2,1:nsteps/2))
shading interp

• How do you calculate the x and y-axis values? Hint: You need to check fft2 function online or check the textbook where it presents FFT with Matlab (Sections 4.4.2 and 4.4.3)
• Compare the numerical dispersion relations obtained experimentally with the two methods and with time steps 1) and 2).
  • What is the impact of the two techniques on the numerical dispersion relation if any? 
  • What is the impact of the Courant number on the numerical dispersion relation if any?