It is possible to simplify the conventional ADI method into Fundamental ADI method, which only has the similar operators at the left-hand sides while being operator-free at the right-hand sides. This may be regarded as the fundamental (basic) scheme 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=https://www.ntu.edu.sg/home/eeltan/papers/2007%20Efficient%20Algorithm%20for%20the%20Unconditionally%20Stable%203-D%20ADI–FDTD%20Method.pdf|journal=IEEE Microwave and Wireless Components Letters|volume=17|issue=1|pages=7-97–9|doi=10.1109/LMWC.2006.887239|via=}}</ref><ref name=":8">{{Cite journal|last=Tan|first=E. L.|date=2008|title=Fundamental Schemes for Efficient Unconditionally Stable Implicit Finite-Difference Time-Domain Methods|url=https://www.ntu.edu.sg/home/eeltan/papers/2008%20Fundamental%20Schemes%20for%20Efficient%20Unconditionally%20Stable%20Implicit%20Finite-Difference%20Time-Domain%20Methods.pdf|journal=IEEE Transactions on Antennas and Propagation|volume=56|issue=1|pages=170-177170–177|doi=10.1109/TAP.2007.913089|via=}}</ref> with no more operator (to be reduced) at the right-hand sides, unlike most traditional 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 fundamental implicit schemes with operator-free right-hand sides.<ref name=":8" /> In their fundamental forms, the FADI method of second-order temporal accuracy can be related closely to the fundamental locally one-dimensional (FLOD) method, which can be upgraded to second-order temporal accuracy, such as 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’sMaxwell's Equations|url=https://www.ntu.edu.sg/home/eeltan/papers/2007%20Unconditionally%20Stable%20LOD-FDTD%20Method%20for%203-D%20Maxwell’s%20Equations.pdf|journal=IEEE Microwave and Wireless Components Letters|volume=17|issue=2|pages=85-8785–87|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=https://www.ntu.edu.sg/home/eeltan/papers/2013%20Unconditionally%20Stable%20Fundamental%20LOD-FDTD%20Method%20With%20Second-Order%20Temporal%20Accuracy%20and%20Complying%20Divergence.pdf|journal=IEEE Transactions on Antennas and Propagation|volume=61|issue=5|pages=2630-26382630–2638|doi=10.1109/TAP.2013.2242036|via=}}</ref> in [[computational electromagnetics]]. For two- and three-dimensional heat conduction and diffusion equations, both FADI and FLOD methods may be implemented in simpler, more efficient and stable manner compared to their conventional methods. <ref>{{Cite journal|last=Tay|first=W. C.|last2=Tan|first2=E. L.|last3=Heh|first3=D. Y.|date=2014|title=Fundamental Locally One-Dimensional Method for 3-D Thermal Simulation|url=|journal=IEICE Transactions on Electronics|volume=E-97-C|issue=7|pages=636-644636–644|doi=10.1587/transele.E97.C.636|via=}}</ref><ref>{{Cite journal|last=Heh|first=D. Y.|last2=Tan|first2=E. L.|last3=Tay|first3=W. C.|date=2016|title=Fast Alternating Direction Implicit Method for Efficient Transient Thermal Simulation of Integrated Circuits|url=|journal=International Journal of Numerical Modelling: Electronic Networks, Devices and Fields|volume=29|issue=1|pages=93-10893–108|doi=10.1002/jnm.2049|via=}}</ref>
