TR-BDF2 Timestepper

[Thomas Bendall (tommbendall)]

PR Summary

Sci/Tech Reviewer: Alex Brown (@atb1995)
Code Reviewer: Pierre Siddall (@Pierre-siddall)

This PR adds a new timestepping scheme called TR-BDF2. Unlike the Semi-Implicit scheme currently used in Gungho, TR-BDF2 consists of two stages: TR (Trapezoidal) and BDF2 (Second-Order Backwards-Difference).
However both stages involve a Quasi-Newton outer/inner iteration like the Semi-Implicit scheme.

Because it contains two stages, the intention is that TR-BDF2 is used with a doubled timestep length. Physics schemes are called only once per the doubled timestep, reducing their cost. It is also possible to reduce the total number of inner iterations compared with the Semi-Implicit scheme, reducing the cost of the linear solver.

TR-BDF2 avoids the implicit off-centering of the Semi-Implicit scheme, which also improves the overall accuracy.

While this scheme will not be immediately adopted, it may be included in GC7 which is why I am lodging the code now.

The changes involved in this PR are:

• introducing namelist settings for the new timestepper
• renaming the inner_iterations setting to inner_iterations_si, and making
  this an array (so that it is possible to perform different numbers of inner
  iteration on each outer iteration)
• passing the stepper enumerator through to the transport scheme, as this the
  TR and BDF steps require different timestep lengths
• setting the tracer transport for TR-BDF2 to happen only in the BDF2 step
• duplicating the dynamics W2 mass matrices. These contain the timestep, and
  different matrices are needed for the TR and BDF2 steps.
• ensuring that the stepper enumerator is available through the argument list
  for any algorithm that calls the dynamics W2 mass matrices, such as rhs_alg
• adding the TR-BDF2 timestepper in all its glory, with physics schemes and diagnostics
• new tests in gungho_model and lfric_atm which use TR-BDF2
• adding documentation(!) of all of the timestepping schemes

Is Blocked By #659

Key Diffs

• As this PR is blocked by Factor timestep out of solver operators #659, the branch has been built upon the branch for Factor timestep out of solver operators #659.
  Therefore the appropriate diff for this PR can be found in this draft PR in my own
  fork: https://github.com/tommbendall/lfric_apps/pull/18/changes
• The action of the upgrade macro and the KGOs can be found on the test branch here: https://github.com/tommbendall/lfric_apps/pull/19/changes

Documentation

• Plots from the new gungho_model and lfric_atm tests, compared with their Semi-Implicit
  equivalents can be found in this PDF:
  results_tr_bdf2.pdf
• The maths of the scheme is documented in the repository's documentation, and I can
  provide reviewers with a path to view the compiled documentation through a browser

Code Quality Checklist

• I have performed a self-review of my own code
• My code follows the project's style guidelines
• Comments have been included that aid understanding and enhance the readability of the code
• My changes generate no new warnings
• All automated checks in the CI pipeline have completed successfully

Testing

• I have tested this change locally, using the LFRic Apps rose-stem suite
• If any tests fail (rose-stem or CI) the reason is understood and acceptable (e.g. kgo changes)
• I have added tests to cover new functionality as appropriate (e.g. system tests, unit tests, etc.)
• Any new tests have been assigned an appropriate amount of compute resource and have been allocated to an appropriate testing group (i.e. the developer tests are for jobs which use a small amount of compute resource and complete in a matter of minutes)

trac.log

Test Suite Results - lfric_apps - test_tr_bdf2/run5

Suite Information

Item	Value
Suite Name	test_tr_bdf2/run5
Suite User	thomas.bendall
Workflow Start	2026-08-17T10:03:08
Groups Run	all', 'lfric_atm_weekly

Dependency	Reference	Main Like
casim	MetOffice/casim@2026.07.1	True
jules	MetOffice/jules@2026.07.1	True
lfric_apps	tommbendall/lfric_apps@test_tr_bdf2	False
lfric_core	tommbendall/lfric_core@932de27	True
moci	MetOffice/moci@2026.07.1	True
SimSys_Scripts	MetOffice/SimSys_Scripts@2026.07.1	True
socrates	MetOffice/socrates@2026.07.1	True
socrates-spectral	MetOffice/socrates-spectral@2026.07.1	True
ukca	MetOffice/ukca@2026.07.1	True

Task Information

✅ succeeded tasks - 1714

Security Considerations

• I have reviewed my changes for potential security issues
• Sensitive data is properly handled (if applicable)
• Authentication and authorisation are properly implemented (if applicable)

Performance Impact

• Performance of the code has been considered and, if applicable, suitable performance measurements have been conducted

AI Assistance and Attribution

• Some of the content of this change has been produced with the assistance of Generative AI tool name (e.g., Met Office Github Copilot Enterprise, Github Copilot Personal, ChatGPT GPT-4, etc) and I have followed the Simulation Systems AI policy (including attribution labels)

AI has been used to help draft the documentation.

Documentation

• Where appropriate I have updated documentation related to this change and confirmed that it builds correctly

PSyclone Approval

• If you have edited any PSyclone-related code (e.g. PSyKAl-lite, Kernel interface, optimisation scripts, LFRic data structure code) then please contact the TCD Team

Sci/Tech Review

• I understand this area of code and the changes being added
• The proposed changes correspond to the pull request description
• Documentation is sufficient (do documentation papers need updating)
• Sufficient testing has been completed

(Please alert the code reviewer via a tag when you have approved the SR)

Code Review

• All dependencies have been resolved
• Related Issues have been properly linked and addressed
• CLA compliance has been confirmed
• Code quality standards have been met
• Tests are adequate and have passed
• Documentation is complete and accurate
• Security considerations have been addressed
• Performance impact is acceptable

[thomasmelvin]

Looks good, i've made a few minor comments, mostly about the descriptions of namelist variables and the timestepping scheme (thanks for doing these!)

A more general question is could we get our existing siqn scheme from careful selection of choices in the TR-BDF2 scheme, i.e. making so that the step for the tr scheme is dt and for the bf2 to is 0. I think you default to the centred siqn scheme but maybe is possible a future change could be to unify these two schemes?

[thomasmelvin]

Is this correct? Most of the gungho orographic tests would run with alpha = 0.5 without any problems. I'm not sure we have tried physics runs with alpha 0.5. I would suggest getting rid of the last sentance

[Thomas Bendall (tommbendall)]

Thanks Tom, I was obviously confused about this point! Shallow water systems can suffer from orographic resonance with the centred scheme, which builds noise that can lead to an instability -- and I had assumed the compressible Euler equations did too. Out of interest I have now tried running the lfric_atm test-suite with alpha=0.5, and every job did actually complete!

I have removed this sentence and also edited the documentation

[thomasmelvin]

I'm not sure this is quite right, after all we also transport u in the outer loop but don't zero the momentum residual in the inner loop. I would say that the reason we zero the density and potential temperature residuals in the inner loop is that the solver updates the transport tendency for these terms (that are only updated in the inner loop) compared to the momentum equation where the solver updates the nonlinear forcing terms (principally the pressure gradient) that are updated in the inner loop

[Thomas Bendall (tommbendall)]

Somehow this got garbled between edits, thanks for pointing it out. I've tried to improve the explanation

[DanStoneMO]

This will require a linked-JEDI PR. I've made changes at: https://github.com/MetOffice/lfric-jedi/pull/1332 though I've not been able to test them yet due to this PR's branch conflicting with main.

I'll approve once this is up-to-date and I can properly test.

[github-actions]

⚠️ Hello Thomas Bendall (@tommbendall)!

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Please do not edit the CONTRIBUTORS.md file. If you have already signed the CLA, revert changes to the file and your signature will be picked up.