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I'm stuck implementing a reverse time migration technique, the PDE(partial differential equation) being implemented is an acoustic wave equation, The numerical method being used is a FD(finite difference method) using SBP(summation by parts) property to ensure stability across the domain,i.e., this changes the coefficients that the FD use in the boundary and near boundary points, in addition to the SAT(simultaneous approximation terms) that will mimic how the continuous energy method is implemented and that would guarantee long time stability across the model. i need help with implementing the SAT for a 2D grid, each time i run the model with the SAT I've implemented it diverges at the last time-steps. IS there any well known acoustic-2D-wave equation that includes SAT to weakly enforce boundary conditions ??

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  • $\begingroup$ You can consider asking a similar (but not identical) question in Scientific Computing SE as well. There are about 175 SE sites and sometimes it's hard to choose the best one. We should not cross-post identical questions in multiple sites, however different-but-related questions are fine, and you can even add a link to each question into the other question if you think it is helpful. Welcome to Stack Exchange! $\endgroup$ – uhoh May 21 at 4:22
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I have no experience with SBP-SAT methods, but you may try and go through the references in (and potentially contact the authors of) this paper https://arxiv.org/pdf/1802.06123.pdf .

Personally, I don't see a strict need for using such 'complicated' modelling methods for RTM. In the ideal case, you have deghosted (i.e., upgoing) recorded data and a smooth velocity model. Then, using standard low-order finite-difference modelling on a uniform grid should already provide fairly good results at fairly little cost. Likely, you'll achieve better results by optimizing the data & velocity model than improving your numerical code beyond the standards. This simple MATLAB code https://csim.kaust.edu.sa/files/ErSE328.2013/LAB/WE_LABS/index.html already does a brilliant job using 2nd order accuracy in time and 8th order accuracy in space, using a simple FD method.

Of course, if you are doing state-of-the-art (academic) research or are dealing with a very atypical velocity model with an extreme dynamic range, these are interesting topics to look into, and I hope that someone may be able to help you... But I hope that my answer may help you right now.

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  • $\begingroup$ Erik, i really want to thank you for reaching out and answering the question, i have much difficulty in expressing the work i'm doing and finding people who can relate. I completely understand your points, however, the main edge that this framework(SBP-SAT) adds to the RTM is that we can use it for a multi-grid technique. where each block can have its own spatial and temporal spacing without damaging the overall stability of the domain and numerical method. This helps a lot in complex velocity domain which can achieve tremendous gain in computational aspect while preserving accuracy. $\endgroup$ – Karim Morad May 20 at 22:05
  • $\begingroup$ Karim, then I think you're best off contacting the authors of related papers such as the one above. Other authors must've experienced most of the typical bugs one could experience when modelling the algorithm. And if you're lucky they may have some sharable code..! $\endgroup$ – Erik May 21 at 9:53

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