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For an undergraduate project I am analyzing the performance of a couple of Crank-Nicolson schemes on a time-dependent problem. For some reason, the order of convergence/general performance of one scheme is noticeably worse when I use an odd number of iterations. I am trying to understand why this is occurring. I am deliberately refraining from sharing the equation for now.

I haven't been able to find any similar issue on the internet except for a couple of possible hits. One paper (Jain, Ray, Bhavsar 2015) had something that sounded like it could be related in one paragraph:

The main idea is that after applying 2 successive Bi-CG steps, it is relatively easy to minimize the residual over that particular Krylov subspace. As a result, the Hybrid BiCGStab method leads only to significant residuals in the even-numbered steps and the odd-numbered steps do not lead necessarily to useful approximations.

Unfortunately, I am not yet at the level of mathematics of the paper and was completely lost.

Has anyone encountered such an issue? If so, where can I read and learn about it? I feel that I may simply be lacking the right terminology or keywords in my search. If no one has ever heard of this, I must assume that there is something wrong with my application of the Crank-Nicolson method on the equation, although I have no idea what it could be if it works half the time.

J. W. Tanner
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D.Bar
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    Could you provide the ODE/PDE you're trying to solve? – VoB Dec 15 '20 at 15:49
  • Can you provide the plot of accuracy as function of number of iterations? Based on your description, it should look like a zigzag, where it gets worse on every odd iteration and better on every even iteration. Is this indeed the case? Have you tried many different iteration numbers? This error may be indicative of a typo in the discretization scheme – Aleksejs Fomins Dec 15 '20 at 17:03
  • @AleksejsFomins Unfortunately I cannot yet provide the ODE. The plot is exactly as you described. I have tried up to 20 iterations. – D.Bar Dec 15 '20 at 19:38
  • Is this with constant time and space step, or is the time interval constant and the space step adapted, or are time and space steps kept proportional? – Lutz Lehmann Dec 15 '20 at 21:58
  • @D.Bar Are 20 iterations supposed to be sufficient for your problem? Did you choose this yourself, or was this number recommended to you? What happens for larger number of iterations? – Aleksejs Fomins Dec 15 '20 at 22:35
  • @LutzLehmann The time step doesn't change at all (no space dimensions). Everything is kept constant- the only thing that changes is the number of iterations. – D.Bar Dec 16 '20 at 11:21
  • @AleksejsFomins 20 iterations are supposed to be more than enough for my problem. Using a larger number of iterations changes nothing (I used 100 to double check). It just keeps zigzagging between the same two points. – D.Bar Dec 16 '20 at 11:32
  • Crank-Nicolson is the combination of the implicit trapezoidal method in time and a symmetric divided difference in space for parabolic PDE similar to the heat equation. What you are describing sounds like the implicit trapezoidal method applied to a high-dimensional system with an iterative sparse solver for the implicit step. What are the 20 iterations related to, the sparse solver or the time integration? How do you measure the accuracy of the method, do you have an exact solution like in the method of manufactured solutions (MMS)? – Lutz Lehmann Dec 16 '20 at 11:42
  • Another association is the power iteration for a matrix with a complex conjugate pair of largest eigenvalues. This iteration can oscillate between the real and imaginary part of an eigenvector, this would look like the described behavior. – Lutz Lehmann Dec 16 '20 at 11:46
  • @LutzLehmann Unfortunately I'm unfamiliar with a few parts of what you've written, so I will have to do some reading and will get back to you once I understand the material you've presented. – D.Bar Dec 16 '20 at 13:33

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