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=== Generalizations ===
The usage of the ADI method as an [[operator splitting]] scheme can be generalized. That is, we may consider general evolution equations
: <math> \dot u = F_1 u + F_2 u, </math>
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>.
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