It is possible to simplify the conventional ADI method into Fundamental ADI method, which only has the same operators at the left-hand sides while being operator-free at the right-hand sides. This may be regarded as the fundamental (basic) implicitscheme of ADI method,<ref>{{Cite journal|last=Tan|first=E. L.|date=2007|title=Efficient Algorithm for the Unconditionally Stable 3-D ADI-FDTD Method|url=|journal=IEEE Microwave and Wireless Components Letters|volume=17|issue=1|pages=7-9|doi=10.1109/LMWC.2006.887239|via=}}</ref><ref name=":8">{{CitationCite journal|last1last=Tan|first1first=E. L.|date=2008|title=Fundamental Schemes for Efficient Unconditionally Stable Implicit Finite-Difference Time-Domain Methods|url=https://www.researchgate.net/profile/E_Tan/publication/3019535_Fundamental_Schemes_for_Efficient_Unconditionally_Stable_Implicit_Finite-Difference_Time-Domain_Methods/links/5ded030b299bf10bc34b0302/Fundamental-Schemes-for-Efficient-Unconditionally-Stable-Implicit-Finite-Difference-Time-Domain-Methods.pdf|journal=IEEE Transactions on Antennas and Propagation|volume=56|issue=1|pages=170–177|year=2008170-177|doi=10.1109/TAP.2007.913089|via=}}</ref> , unlike common othertraditional implicit methods that usually consist of operators at both sides of equations. The FADI method leads to simpler, more concise and efficient update equations without degrading the accuracy of conventional ADI method.
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=== Relations to other implicit methods ===
Many classical implicit methods by Peachman-Rachford, Douglas-Gunn, D'Yakonov, Beam-Warming, Crank-Nicolson, etc., may be simplified to FADI method with operator-free right-hand sides.<ref name=":8" /> In their respective fundamental implicit formsschemes (with operator-free right-hand sides), the FADI method of second-order temporal accuracy can be related closely to the fundamental locally one-dimensional (FLOD) method that is also of second-order temporal accuracy, for three-dimensional Maxwell's equations. <ref>{{Cite journal|last=Tan|first=E. L.|date=2007|title=Unconditionally Stable LOD-FDTD Method for 3-D Maxwell’s Equations|url=https://www.researchgate.net/profile/E_Tan/publication/3429376_Unconditionally_stable_LOD-FDTD_method_for_3-D_Maxwell's_equations/links/5ded0d804585159aa46e6f46/Unconditionally-stable-LOD-FDTD-method-for-3-D-Maxwells-equations.pdf|journal=IEEE Microwave and Wireless Components Letters|volume=17|issue=2|pages=85-87|doi=10.1109/LMWC.2006.890166|via=}}</ref><ref>{{Cite journal|last=Gan|first=T. H.|last2=Tan|first2=E. L.|date=2013|title=Unconditionally Stable Fundamental LOD-FDTD Method with Second-Order Temporal Accuracy and Complying Divergence|url=|journal=IEEE Transactions on Antennas and Propagation|volume=61|issue=5|pages=2630-2638|doi=10.1109/TAP.2013.2242036|via=}}</ref>
