examples: Add an example notebook for ADER-FD schemes #2338
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examples/misc/01_ader_fd.ipynb
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Yeah, still need to add these
examples/misc/01_ader_fd.ipynb
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Line #10. # dv.grad(dv.div(v)) does not get flattened and will need to be explicitly written out
Not sure what that means but does not make sense to have to go to those sympy matrices.
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Maybe I'm overthinking? I was under the impression that dv.grad(dv.div(v)) is going to result in bigger stencils than doing the grad of the divergence algebraically and implementing that
examples/misc/01_ader_fd.ipynb
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examples/misc/01_ader_fd.ipynb
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Isn't that second order just the OT4 acoustic kernel but for the first order equations?
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examples/misc/01_ader_fd.ipynb
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| "# First time derivatives\n", | ||
| "pdt = rho*c2*dv.div(v)\n", | ||
| "vdt = b*dv.grad(p)\n", |
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This is quite surprising, it's missing the time leap-frogging
p.forward = f(v, p))
v.forward=f(p.forward, v)
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Yeah, you don't need time leapfrogging for ADER. You also don't need any staggering to solve first-order systems of equations. This is part of the appeal imo on top of high-order timestepping
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What's the reason this is currently located in misc vs the seismic tutorials? Locating in the later and referring previous tutorials where appropriate would seem more suitable to me? |
examples/misc/01_ader_fd.ipynb
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I would recommend either defining the state variables or referencing previous tutorials appropriately. Unless this tutorial is only intended for readers who are familiar with seismic modelling, make it as simple for target audience as is reasonably possible - that is, make sure everything is clearly defined.
I'm not a big fan of the phrasing here:
"one can derive expressions for these higher-order time derivatives in terms of spatial derivatives. High-order..."
Just say what you're going to do in the tutorial and reference the paper sited above as appropriate.
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Also, U expansion expression is missing
examples/misc/01_ader_fd.ipynb
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examples/misc/01_ader_fd.ipynb
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No need for "as usual". Ref. the difference of this setup vs the one in the existing first order notebook.
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examples/misc/01_ader_fd.ipynb
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examples/misc/01_ader_fd.ipynb
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Conventional case? Conventional to whom?
"Note that if one assumes non-constant material parameters when deriving the higher time derivatives, it will be necessary..." - reference or brief explanation.
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examples/misc/01_ader_fd.ipynb
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Don't we define the biharmonic method in differentiable?
Not necessarily for this PR, but with a couple of appropriate sympy functions deriving expressions of arbitrary order could be 'automated'.
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examples/misc/01_ader_fd.ipynb
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Add labels etc.
Would be nice to add a comparison vs staggered 1st order acoustic (with a difference plot).
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This was useful, since it turns out this highlights the higher stability limit of ADER vs staggered
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Still need to check the mathematical details in some places. But in general, throughout the tutorial recommend expanding and clarifying statements a little more and adding a comparison vs staggered acoustic. |
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Because Devito derivatives are discrete, so grad(div()) is the gradient stencil applied to a divergence stencil. This will contain f[0].dx.dx which is not the same stencil as f[0].dx2
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Line #7. # Update equations (3rd-order ADER timestepping)
I would prefer writing the PDE and use solve rather than by hand. It would also make the method more explicit
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That would obfuscate how the timestepping works imo
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No. If you imposed all the higher-order conditions to get a free surface, you would expect it to reflect, but the Dirichlet boundary condition when you hit the zero padding isn't really a free surface or anything physically meaningful. There are reflections back into the domain in the ADER case, they're just much lower amplitude. Due to the properties of the discretisation and the fact that the padding is held at zero, energy is diffused at the edges of the domain, hence the lack of reflections
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Mostly minor comments. Neess reabase as well. |
Currently work in progress. Adds a notebook showing how to solve the first-order formulation of the acoustic wave equation with 4th-order ADER timestepping.
Lmk your thoughts.