The system of equations involved is [[symmetric matrix|symmetric]] and tridiagonal (banded with bandwidth 3), and is typically solved using [[tridiagonal matrix algorithm]].
It can be shown that this method is unconditionally stable and second order in time and space.<ref>{{Citation | last1=Douglas, Jr. | first1=J. Jr.| title=On the numerical integration of u<sub>xx</sub>+ u<sub>yy</sub>= u<sub>t</sub> by implicit methods | mr=0071875 | year=1955 | journal=Journal of the Society for Industrial and Applied Mathematics | volume=3 | pages=42–65}}.
</ref> There are more refined ADI methods such as the methods of Douglas,<ref>{{Citation | last1=Douglas Jr. | first1=Jim Jr.| title=Alternating direction methods for three space variables | doi=10.1007/BF01386295 | year=1962 | journal=Numerische Mathematik | issn=0029-599X | volume=4 | issue=1 | pages=41–63| s2cid=121455963 }}.</ref> or the f-factor method<ref>{{Citation | last1=Chang | first1=M. J. | last2=Chow | first2=L. C. | last3=Chang | first3=W. S. | title=Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems | doi=10.1080/10407799108944957 | year=1991 | journal=Numerical Heat Transfer, Part B: Fundamentals | issn=1040-7790 | volume=19 | issue=1 | pages=69–84| bibcode=1991NHTB...19...69C | url=https://zenodo.org/record/1234469 }}.</ref> which can be used for three or more dimensions.
