There are three main approaches to discretize Maxwell equations: finite differences, finite elements and finite volumes. However, naïve implementations of these finite methods to discretize Maxwell equations on irregular grids are known to cause problems, such as spurious modes and late-time unconditional instabilities, that destroy the solutions. This is often a consequence of the failure to capture some essential physics of the continuum theory. Compatible discretizations have been proposed to inherit and mimic those essential physical structures of Maxwell equations.
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