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

[iboutle]

Just to note that care will need taking merging this with #709 and #734

[thomasmelvin]

These changes all look good to me, thanks for updating the documentation

[Alex Brown (atb1995)]

I think a summary table would be very useful here. In particular, equation set being solved (Transport vs Compressible Euler), and specifically for TR-BDF2 the standard number of calls to the main components, i.e. linear solver, physics etc.

[Alex Brown (atb1995)]

Or some sort of headline description of each of the methods

[Thomas Bendall (tommbendall)]

Thanks for the suggestion I've gone for the headline description -- the table for comparing number of calls is really useful but depends on timestep and configuration, which at the moment is a bit of a guess!

[Alex Brown (atb1995)]

This ticket adds a new timestepper - the SISL version of TR-BDF2. It make use of much of the infrastructure that exists for SIQN, but refactors many of the common components. As noted by TB, TR-BDF2 is a two stage implicit method, which maintains second order accuracy (and L-Stability) without off-centering and reduction in order.

The PR/changes are thorough and well written, and the addition of the timestepping docs is a very welcome one.

I only have a small number of commends (already added to the PR). I realise this is the first step before a thorough analysis on whether to adopt TR-BDF2 instead of SIQN. In longer term (not part of this PR) I would suggest a handful of investigations:

1. SBR test case - much deeper, open an issue to investigate this
2. I am sure an investigation will be done in GMED into TR-BDF2 performance for global NWP and climate modelling, and that work has been done looking at TR-BDF2 in Gusto for dycore only problems. But a direct comparison (TR-BDF2 vs SIQN) using GungHo for a variety of dycore test cases would be very welcome

[Thomas Bendall (tommbendall)]

This ticket adds a new timestepper - the SISL version of TR-BDF2. It make use of much of the infrastructure that exists for SIQN, but refactors many of the common components. As noted by TB, TR-BDF2 is a two stage implicit method, which maintains second order accuracy (and L-Stability) without off-centering and reduction in order.

The PR/changes are thorough and well written, and the addition of the timestepping docs is a very welcome one.

I only have a small number of commends (already added to the PR). I realise this is the first step before a thorough analysis on whether to adopt TR-BDF2 instead of SIQN. In longer term (not part of this PR) I would suggest a handful of investigations:

1. SBR test case - much deeper, open an issue to investigate this
2. I am sure an investigation will be done in GMED into TR-BDF2 performance for global NWP and climate modelling, and that work has been done looking at TR-BDF2 in Gusto for dycore only problems. But a direct comparison (TR-BDF2 vs SIQN) using GungHo for a variety of dycore test cases would be very welcome

Thanks so much for the review. I think I've responded to your points.

I agree about your suggested investigations. I haven't opened an issue since TR-BDF2 is still only a research option, but these are on my own list of things to do imminently.

[Alex Brown (atb1995)]

Thanks Tom. This looks good to me, happy to approve

[Thomas Bendall (tommbendall)]

Pierre Siddall (@Pierre-siddall) this is ready for code review

[DrTVockerodtMO]

Thanks Tom. Sorry to be a pain but some of the adjoint routines need the stepper enum included in the interface. Part of this is preference to have things matched up (even if it goes unused), but one of the adjoint routines is missing the updated dt_rtran logic.

[github-actions]

⚠️ Hello Thomas Bendall (@tommbendall)!

Your CLA signature was found on the base branch, but you appear to have modified the CONTRIBUTORS.md file in this PR.

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.

[DrTVockerodtMO]

Thanks for making those changes, LGTM!