Add Deformational Flow Example

[dennisYatunin]

Purpose

This PR was intended to demonstrate the correct convergence behavior of SSP methods with limiters, using the Deformational Flow test case from the DCMIP test case document. Unfortunately, it appears that neither ARK nor SSP algorithms converge when a non-trivial limiter is applied.

Closes #162

To-do

Determine whether the observed behavior is expected, or if there is a bug in either the test setup or the SSP implementation.

Content

The Deformational Flow test case has been ported over from ClimaAtmos, but simplified to more closely resemble the description in the DCMIP document. Specifically, density has been removed as a prognostic variable, and the equations have been converted to advective form. The test case can be run with three "limiter modes"---no limiter, the limiter used in ClimaCore, and a custom operation. The last option is helpful for testing out various simple alternatives, and it is currently set to rescale each tracer concentration field so that the total sum of the tracer mass remains constant.

Whenever the test case is run with no limiter, the correct convergence behavior is observed; this can be verified through the convergence order plot for this test case in the docs (located in Developer Docs/Report Generator). While this PR is in the testing phase, the plot only includes SSP333 and ARS343. Whenever a limiter is applied, the convergence order drops; it seems to drop to around 1 for very small values of dt with both current limiter options, but further testing is needed to be certain of this.

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• I have read and checked the items on the review checklist.

[dennisYatunin]

Here are the plots for the convergence of SSP333 and ARS343 without any limiter and with the limiter from ClimaCore. Since the test case is currently fully explicit, we expect SSP333 to converge with order 3 and ARS343 to converge with order 4. The analytic solution is only defined at the start and end times, so the plots only show the solution/error norms at those times (even though the errors are computed against a numerical reference solution, that reference is still checked against the analytic solution).

[tapios]

This is the expected behavior (Godunov's theorem)

[dennisYatunin]

This PR has been updated to insert an analysis of IMEXAlgorithm with limiters into the documentation, recreating Table 1 from "Optimization-based limiters for the spectral element method" by Guba et al., along with plots of all the values used to generate Table 1. Should be good to merge once the documentation build succeeds.

[bischtob]

@dennisYatunin are we merging this by EOW?

[dennisYatunin]

bors r+

[bors]

Build failed:

• format

[dennisYatunin]

bors r+

[bors]

Build succeeded:

• buildkite/climatimesteppers-ci
• docbuild
• format
• test-os (1.8, macos-latest)
• test-os (1.8, ubuntu-latest)
• test-os (1.8, windows-latest)