Excerpt from Alternating Direction and Semi-Explicit Difference Methods for Parabolic Partial Differential Equations

For the model problem, the first boundary value problem for the heat conduction equation in a rectangular domain, the unconditional stability of the alternating direction methods was proved in [3] and The proof consists in showing, with the aid of Fourier analysis, that the von Neumann stability condition [11] is always satisfied. It can be shown however, that this method of proof cannot be extended beyond the model problem.

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