I'm using the Crank-Nicholson method to solve the time-dependent Schrödinger equation with the split-operator method. I'm getting some weird results that are probably the result of a bug somewhere in my code.

Just in case, I thought I should probably check to see if the method I'm using is unstable.

Does the split-operator method change the stability properties of the Crank-Nicolson method?

If so, how?

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    $\begingroup$ I haven't analyzed or experimented with this specific case, but certainly the use of ADI can in general affect the stable timestep. However, in the case of the implicit trapezoidal rule, I wouldn't expect any change as it is A-stable. $\endgroup$ – David Ketcheson Feb 13 '12 at 6:40
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    $\begingroup$ @David Ketcheson: You may want to write that up as an answer. $\endgroup$ – Dan Feb 16 '12 at 17:49

The time-dependent Schrödinger equation is not really a heat equation. Still, the Crank–Nicolson method is well suited for its solution. However, the Crank-Nicolson method is fully implicit, so the statement "doing the implicit part with ADI" sounds a bit suspicious. It probably means that the diffusion like part is done with ADI. I wonder a bit whether that means the potential part is treated by an analytical solution together with another application of an operator-split scheme. (But why would we call this a Crank-Nicolson method?)

For the heat equation, normally the ADI methods Peaceman–Rachford, Douglas–Rachford and Douglas–Gunn get discussed. I'm not so sure how much this analysis carries over to fake/formal heat equations, but at least Douglas–Rachford is certainly unsuitable for the Schrödinger equation. There certainly are stable ADI schemes that can be used for the Schrödinger equations (probably Douglas–Gunn works), but an arbitrary ADI scheme that works well for the heat equation is not guaranteed to also work well for the Schrödinger equation. But even if it would be unstable, it would probably be only weakly unstable, so that you should still be able to get "some" results. So really "weird" results probably have a different origin than the stability of the ADI scheme.

  • $\begingroup$ "Implicit part" was just poor wording. I was using Crank-Nicholson steps on dimension. I was using the Baker-Campbell-Hausdorf expansion to get the particular form that I used. $\endgroup$ – Dan Feb 1 '13 at 21:51
  • $\begingroup$ @Dan Because you reference the Baker-Campbell-Hausdorf expansion, I guess you do exactly what I called "application of an operator-split scheme". I guess it's a Strang splitting. When I google for "Strang splitting", the first hit contains "time-dependent Schrödinger", "Baker-Campbell-Hausdorff formula" and also introduces the Strang splitting... $\endgroup$ – Thomas Klimpel Feb 1 '13 at 22:28

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