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APPROACH TO NUMERICAL SOLUTION OF EM PROBLEMS 453

�(∆) + (∆) + (∆) as ∆ ≤ . This limit is necessary but not sufficient to guarantee
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2
2
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stability. Roughly speaking, it means that the time step must be kept small enough so that
information has enough time to propagate through the space discretization. There are several
variants of FDTD that are free from this restriction but at the cost of more complex code with
more bug probability.
• Can handle up to hundred million cells’ problem giving a solution in minutes on high-
performance computers allowing the simultaneous use of multiple computer resources (for
example, the distributed multi-processor computing with effective resource management).
However, the mere increase in the number of processors above some limit is not justified. More
processors mean the greater time waste on the data exchange between processors that can slow
down the simulation procedure. Meantime, the simulations to be completed over 35 times faster
if the computer is equipped with Graphical Processing Unit (GPU) processor, which is
specifically designed to handle large amounts of graphical data in parallel. A modern GPU has
several hundred small processors that can work in parallel. The experiment demonstrates that
GPU processor combined with CPU cache memory usage may accelerate simulations from
1500 to 21000 Mcells/s compared to non-accelerated standard multi-core workstation speeds
ranging from 20 to 200 Mcells/s. One of such ultra-fast and powerful commercial FDTD tools
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is EMPIRE XPU .
• More than a dozen specific and general purpose commercial simulators are available on the
market.

The primary drawbacks of FDTD technique are:
• Since FDTD requires that the entire computational domain is gridded, and the grid spatial
discretization must be sufficiently fine to resolve both the shortest EM wavelength and the
smallest geometrical feature in the model. The consequence is large computational domains
and relatively long solution times. In general, an FDTD requires 30 bytes of memory per Yee
cell [3]. To estimate the total memory required, in bytes, just multiply the whole number of
FDTD cells by 30. There is some overhead in the calculation, but it is generally quite small.
• Run times in the order of hours, days, or even longer are common when solving
electromagnetic waves problems of practical size. The simplest way to estimate this time is to
multiply the total number of cells by the expected time steps and the factor 80 that describes
the required amount of operations per cell and per time step. If a time duration of each floating-
point operation is known, the whole execution time on a single processor can be projected. In
general, though, a better estimating method is to determine the execution time of a simple
problem on a given computer and then scale the time by the ratio of the number of operations
between the desired calculation and the simple one.
• Since FDTD simulations calculate the E- and H-fields at all points within the computational
domain, the latter must be finite to permit its residence in the computer memory. As such, the
Perfectly Matched Layer (PML) box is needed to properly truncate the spatial domain in case
of exterior EM problems (see below Section 9.1.3).
• Since the computational grids are typically rectangular, they do not conform nicely curved
surfaces that cause the additional meshing inaccuracies. Irregular, non-orthogonal grids are

11 Courant-Friedrichs-Lewy (CFL) criterion.
12 Check http://www.empire.de/