Stability of the discretized wave equation. Analyze the stability of the algorithm presented in Problem 3.4 using the von Neumann method. (This will require some steps similar to the leapfrog algorithm.) Does the stability criterion derived here agree with the results of Problem 3.4? 5.6. Stability in the FDTD method. Write a 1D FDTD simulation 400 grid cells long for a wavelength of 1 m, with ?x = λ0/20. Launch a Gaussian pulse from the left end of the space with the same parameters as Problem 5.4 above. Now, increase ?t very slightly to excite an instability, similar to Figure 5.4. Can you measure the growth rate of this instability? Does this growth rate agree with q from Equation (5.14)? 5.7. Stability in the 2D FDTD method. Write a 2D TM mode FDTD simulation that is 100 grid cells in the y-dimension and 500 cells in the x-dimension. Use PEC at the far right boundary; at the top and bottom boundaries, set .,
replace the unknown fields with their counterparts on the opposite boundary. This is called a periodic boundary condition, which we will study in Chapter 12. Launch a plane wave from the leftmost boundary at a frequency suitable to the grid cell size. Now, investigate the stability of this problem by varying ?t; how large can you make ?t before the simulation goes unstable? Can you surpass the CFL condition? Why or why not?