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A non-deteriorating algorithm for computational electromagnetism based on quasi-lacunae of Maxwell’s equations
The performance of many well-known methods used for the treatment of outer boundaries
in computational electromagnetism (CEM) may deteriorate over long time intervals. The
methods found susceptible to this undesirable phenomenon include some local low order
artificial boundary conditions (ABCs), as well as perfectly matched layers (PMLs). We pro-
pose a universal algorithm for correcting this problem. It works regardless of either why
the deterioration occurs in each particular instance, or how it actually manifests itself (loss
of accuracy, loss of stability, etc.). Our algorithm relies on the Huygens’ principle in the
generalized form, when a non-zero electrostatic solution can be present behind aft fronts
of the propagating waves, i.e., inside the lacunae of Maxwell’s equations. In this case, we
refer to quasi-lacunae as opposed to conventional lacunae, for which the solution behind
aft fronts is zero. The use of quasi-lacunae allows us to overcome a key constraint of the
previously developed version of our algorithm that was based on genuine lacunae. Namely,
the currents that drive the solution no longer have to be solenoidal. Another important
development is that we apply the methodology to general non-Huygens’ problems.