Domain diagnostics

[nilsvu]

We need to improve our capabilities to analyze where our computational grids need more or less resolution. This is important for many reasons:

• We need to understand if some parts of the domain are severely underresolved, leading to large numerical errors.
• Wasting resolution on regions of the domain that are overresolved significantly increases the cost of a simulation.
• These diagnostics form the basis for adaptive mesh-refinement (AMR) algorithms that adjust the resolution automatically throughout a simulation.

Here's a list of useful diagnostics. We should evaluate the strengths and limitations of these quantities carefully while implementing them. This knowledge will be important for implementing AMR algorithms, so please document your findings!

• [x] Power monitors. See Sec. 5.1 in https://arxiv.org/abs/1405.3693. @nikwit and I have implemented these partially in Python, so talk to us.
• [x] Numerical vs analytic Jacobian (see @nilsdeppe's comments below).

[nilsdeppe]

I think another reasonable diagnostic is to compare the analytic Jacobian compared to the numerical logical partial derivative of the inertial coordinates. Specifically the (relative) difference between the too should give a good indication of how resolved the maps are

[nilsdeppe]

Thinking more about the Jacobian metric, you want to look at

$$ C{\hat{\imath}} = 1 - \frac{\sum{i} \partial{\hat{\imath}} x^i }{\sum{i} D_{\hat{\imath}} x^i}. $$

You want to do the sums in the numerator and denominator to avoid any division by zero. Another alternative is

$$ C{\hat{\imath}} = 1 - \sum{i} \frac{\partial{\hat{\imath}} x^i }{D{\hat{\imath}} x^i + \epsilon} $$

where $\epsilon$ is a parameter to avoid division by zero and should be small (e.g. 1e-20 or so). In both forms $\partial{\hat{\imath}}x^i$ is the analytic Jacobian while $D{\hat{\imath}} x^i$ is the numerical Jacobian computed using logical_partial_derivatives.

[nilsdeppe]

There is also a power monitor/AMR implementation in dg-charm on bitbucket, so when someone starts working on that, please ask to have access to that code.

[guilara]

Hi @nilsdeppe. I've started looking into this. Could you please give me access to the code you are mentioning? Thanks!