FDTD analyses produce solutions that fail to propagate through the cells at the proper phase velocity in all directions. The propagation velocity depends on the cell size in wavelengths; it has a frequency-dependent component. You need to consider this numerical dispersion because it affects accuracy. Because the waves travel at different velocities in different directions, the dispersion problem increases for large structures where many time steps must be taken. After many steps, signals disperse because they have taken different routes and fail to add together in the correct phase. Finer cells solve the problem, but the computation requirements grow rapidly. The equation for the propagation constant can be found from considering the FDTD formulation to produce the following equation for three-dimensional problems:
 (2-54)The factor is the FDTD propagation constant in the cells along the x-axis, only approximately the same as , the actual propagation constant in the structure. The yand z-axes have similar problems. If you take the limit as cell length approaches zero, , and so on, then  . Because  as the cell size shrinks for the solution still to satisfy the Courant limit, Eq. (2-54) reduces to the expression
 (2-55)Equation (2-55) is the normal propagation constant equation for a plane wave in space and shows that the cell propagation constants converge to the correct values as the cell size shrinks. If you formulate a problem in one or two dimensions, you remove terms from the right side of Eq. (2-54) to determine the dispersion relationship.
Absorbing boundary conditions (ABCs) can cause numerical instabilities. ABCs approximate infinite space to simulate radiation by the antenna into space. FDTD problems must be placed in a finite number of cells because each cell requires computer storage. Every FDTD problem uses a finite number of cells for the ABCs with more cells required in the directions of maximum radiation. ABCs degrade as the number of time steps increases and eventually leads to numerical instabilities. A lively research on ABCs has produced good ones, but be aware that most have been found to produce problems at some point. If you write your own analyses, you will need to find appropriate ones. Commercial codes will give their limitations.
At one time, ABCs limited solution dynamic range, but ABCs are now available that produce reflection coefficients from . Numerical dispersion limits the dynamic range as well. Remember that the antenna will be modeled with small cubes that limit the resolution of the results. The errors of modeling lead to solution errors that limit the dynamic range. |
