Add a fully coupled periodic implicit operator

[rois1995]

Proposed Changes

Closed at the author’s request: deferred experimental enhancement. No overall benefit has been established; the branch and benchmark evidence are retained for possible future investigation. The low-linear-budget convergence regression remains unresolved; the preconditioner experiment is unpublished and fails expanded mixed-precision solve checks. Full CFD/geometry adjoint, multigrid, ALE/GCL and regression validation are incomplete.

This PR proposes a fully coupled periodic implicit operator in place of the existing approximate linearization. The omitted neighboring Jacobian terms were a deliberate tradeoff to reduce communication and avoid extending the sparse matrix. A more complete operator is a numerical enhancement; disagreement with its dense reference alone does not establish a bug in that approximation. This PR is classified as changelog:feature, without priority.

Dependencies: #2961 and #2963. Their commits are included until those PRs merge; the focused diff is axis...implicit.

The proposed matrix product includes neighboring contributions from all periodic copies through the constrained operator P A P + I-P, where P averages periodic copies in a common frame. Forward/transpose products and linear-solve reverse callbacks use the same operator; residual averaging and time terms follow that representation. Unsupported CUDA and direct PaStiX solves are rejected rather than silently applying an incomplete operator.

An independent dense reference verifies the proposed operator through products, transpose dot identities and forward/reverse solves for translation, helical rotation and two/three pairs on genuinely partitioned grids. Actual reverse-AD callback sensitivities agree with finite differences. Controlled annular/axis implicit sector/full steps agree below 4e-11 relative.

An isolated comparison now measures the practical effect of this coupling while keeping the other periodic fixes identical. Both modes use identical input, CFL, linear tolerance and iteration cap, and stop at log10(rms[Rho]) <= -10:

Serial case	Outer iterations, legacy → this PR	Total linear iterations, legacy → this PR
45° annulus, CFL100, max50 linear iterations	2337 → 807	5063 → 19818
30° pipe with axis, CFL10, max20 linear iterations	1698 → 1062	6792 → 21240

The PR reduces outer iterations in these cases, but increases total linear work and is slower in the recorded runs. No general speedup is established. With the annulus restricted to four linear iterations per step, legacy converges while this PR stalls through 4000 outer iterations; the regression also occurs with two MPI ranks. Raising the linear iteration cap does not resolve that robustness concern.

Exact cases, histories, timings, convergence plot and reproduction instructions are published in evidence commit 2667a182cf. An unpublished projected-preconditioner experiment reduces linear work in some cases, but still stalls in the low-budget case and fails expanded mixed-precision solve checks. It is not included in this PR; the branch source is unchanged.

Unresolved validation: the low-linear-budget convergence regression and preconditioning assessment remain open. This changes the implicit periodic discretization and can change existing periodic regression references. Full CFD/geometry sensitivities, multigrid forcing, ALE/GCL and full regression/reference updates remain open. The callback check proves RHS sensitivity of the linear solve, not all flow/mesh derivatives. Enabled GPU/PaStiX builds also remain untested.

Validation of the combined periodic source passes serial, partitioned MPI2, OpenMP2 and MPI2×OpenMP2 (10 cases / 3175 serial assertions). Individual branch CI and complete regression/reference checks are pending. Test configurations, meshes, logs and before/after values are in the testcase comment. New regression references are local x86 values and need CI confirmation.

Related Work

• MPI Point-to-Point Refactoring + New Periodic BC Implementation #652 introduced the current periodic implementation.
• Fixes for rotational periodic boundaries: implicit solver, multigrid and limiters #2961 lists the seven active periodic bug-fix PRs and this closed experiment. Their numbering groups the topics; it does not imply a linear dependency chain.
• Instability in BL with Periodic Boundary Conditions #1467 and Periodic boundary conditions and simple timestepping don't combine well #763 discuss periodic convergence/time-stepping concerns; this PR does not yet establish that it resolves those reported cases.
• #1585 explored a similar enhancement and documented increased linear work from inconsistent preconditioning and discrete-adjoint difficulties. The required geometry implementation is in Fix periodic axis geometry, slip normals and moving donor volumes #2963.

PR Checklist

• I am submitting my contribution to the develop branch.
• My contribution generates no new compiler warnings (complete branch CI pending).
• My contribution is commented and consistent with SU2 style.
• I ran the repository pre-commit checks on the changed files.
• I have added tests that demonstrate the contribution.
• I have updated appropriate documentation, if necessary.

[rois1995]

Test cases

Reproducers, configurations, numeric logs and scripts: implicit. Develop = 6db10127d1; BOX fixtures generate their meshes. The original B–D evidence above was recorded before the expanded follow-ups.

Check	develop	fixed combined source
Independent dense constrained operator, 4 layouts	16/2924 assertions fail	passes
Translation forward product / solve	max error 2.4272 / 0.9857	passes
Translation transpose / reverse solve	max error 1.51059 / 0.86296	passes
Helical product / solve / transpose / reverse	3.4644 / 1.40676 / 2.21873 / 4.07183	passes
Controlled implicit sector/full physical step, annulus and axis, extra translation pair	dense operator defect establishes baseline; no baseline sector-flow result claimed	all field-relative differences below 4e-11
Actual external reverse callback vs finite differences	no baseline callback result claimed	translation/helical, 4 assertions/rank, serial and partitioned MPI2

The dense reference is assembled independently from local blocks and global connectivity; MPI tests require owned points on both ranks. Products and transpose checks keep 1e-5 bounds. Float Krylov solution checks use sqrt(float epsilon), with requested residual 1e-7; double uses 1e-10 and 1e-5 solution bounds. An initially double-only float-solve criterion failed and is documented as a testcase precision issue, not hidden as a pass.

The controlled step uses physical dt 1e-4 and zero JST dissipation. Split-face nonlinear dissipation and local pseudo-time conventions can still produce different sector/full results in other setups. Full CFD/geometry AD, multigrid forcing, ALE/GCL, complete regression references and GPU/direct-solver builds remain open; this PR stays a draft.

Combined release checks: serial and OpenMP2 pass 10 cases / 3175 assertions; partitioned MPI2 and MPI2×OpenMP2 pass on both ranks (2334 / 2238 assertions). These are combined-source checks, not standalone builds of every branch. Complete branch CI and full regression/reference checks remain pending.