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 ??
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.