Alternating-direction implicit method: Difference between revisions

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where <math> F_1 </math> and <math> F_2 </math> are (possibly nonlinear) operators defined on a Banach space.<ref>{{cite book|last2=Verwer|first2=Jan|first1=Willem|last1=Hundsdorfer|title=Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations|date=2003|publisher=Springer Berlin Heidelberg|___location=Berlin, Heidelberg|isbn=978-3-662-09017-6}}</ref><ref>{{cite journal|last1=Lions|first1=P. L.|last2=Mercier|first2=B.|title=Splitting Algorithms for the Sum of Two Nonlinear Operators|journal=SIAM Journal on Numerical Analysis|date=December 1979|volume=16|issue=6|pages=964–979|doi=10.1137/0716071|bibcode=1979SJNA...16..964L}}</ref> In the diffusion example above we have <math> F_1 = {\partial^2 \over \partial x^2} </math> and <math> F_2 = {\partial^2 \over \partial y^2} </math>.

== Simplification ofFundamental ADI into (FADI (Fundamental ADI) method ==

=== Simplification of ADI method to Fundamental ADI method ===

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) 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=|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">{{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.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|doi=10.1109/TAP.2007.913089|via=}}</ref> unlike 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 FADIfundamental methodimplicit schemes with operator-free right-hand sides.<ref name=":8" /> In their respective fundamental implicit schemes (with operator-free right-hand sides)forms, the FADI method of second-order temporal accuracy can be related closely to the fundamental locally one-dimensional (FLOD) method, thatwhich 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> in [[computational electromagnetics]].

== References ==