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We have an adaptive mesh refinement (AMR) code for solving the elastic wave equation with frictional fault interfaces (based on Chombo for those that are interested). One of the things that we have realized is that our results are being strongly affected by the presence of the outer absorbing boundary (which we implement as a simple characteristic boundary condition). For reference we currently use the multidimensional Godunov (Finite Volume) scheme of Colella and collaborators. Though we are not wed to these methods (just easy to use since they are already in Chombo) we do need adaptivity in time.

What I am wondering is if anyone one has any experience with more efficient absorbing boundary conditions with AMR using adaptive time stepping, such as perfectly matched layers or high-order boundary conditions. Any reason not to go down this road? My limited searching hasn't really turned up any useful references or mentions of this in the literature.

Edit: clarified that this is a finite volume method.

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  • $\begingroup$ At least for Maxwell solvers, perfectly matched layers are used together with all different sorts of solvers (FDTD, ADI, FEM, time domain, "time harmonic", static, ...). The initial problems (like long time instability, performance for grazing incidence, ...) have been overcome/addressed/solved a long time ago. $\endgroup$ – Thomas Klimpel Feb 4 '12 at 23:03
  • $\begingroup$ I guess I realize that for the continuous problem things have been worked out. But I do know that for linear elasticity some people are/have reported in stability problems with DG and SpecFem methods. So I wasn't sure if AMR would cause any additional problems given the coarsening and refining of auxiliary variables. I will likely just try it and see since it is not too difficult to add to the code. $\endgroup$ – Jeremy Kozdon Feb 5 '12 at 21:33
  • $\begingroup$ I doubt there should be any significant problems with the discretization methods unless it impacts the physics of the model if the modes of interest are sufficiently resolved. For wave absorption zones/layers it may be appropriate to tune to the physics that are resolved for these zones to be efficient (fx. length/size and amount of damping). $\endgroup$ – Allan P. Engsig-Karup Feb 6 '12 at 21:34
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Is this using finite elements?

I dont know much about PMLs but as long as the implementation is local to the element then it should not be a problem.

Afaik PML implementation in the frequency domain is local i.e., elements end up having a modified mass matrix, material coefficient matrix and the strain-displacement matrix. Not sure about the time domain.

You can always use viscous dampers as the implementation is very simple only requiring changes to the element damping matrix.

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  • $\begingroup$ It is a finite volume method within the Berger-Oliger framework. The thing I am wondering about are the coarsening and refinement operations with the auxiliary variables as well as whether there are any known issues with multi-D methods and PMLs. $\endgroup$ – Jeremy Kozdon Feb 4 '12 at 16:58

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