Nothing in this chapter is in this release. Planned Chapters 2 through 9 describe formulations the shipped engine executes; this chapter records the ones it does not — designed, prototyped or measured and set aside, or once shipped and withdrawn — so that a plan file cited in a commit, a retired option key met in an old model, or a limitation stated in an earlier chapter can be followed to its record. Planned means designed and not built, whatever the plan's own optimism; Retired means explored or shipped and withdrawn, with the measurement that withdrew it. Every section names its plan of record by path; the plans are not part of the manual, and the code is the source of truth where they disagree.

Section	Formulation	Plan of record	Status
10.1	Chebyshev spectral geometry for irregular sections	`plans/CHEBYSHEV_XSECT_GEOMETRY_PLAN.md`, `plans/CHEBYSHEV_XSECT_FEASIBILITY_2026-08-21.md`	Planned
10.2	An implicit finite-volume solver	`plans/FV_IMPLICIT_HYPRE_SOLVER_EXPLORATION.md`	Planned
10.3	The 1D finite-volume solver on Kokkos backends	`plans/EXPLICIT_FV_KOKKOS_1D_SOLVER_PLAN.md` §5, §7B.3; `ROADMAP.md` §7	Planned
10.4	Dynamic wave on GPU	`plans/1d/1D_DYNWAVE_GPU_KOKKOS_STRATEGY.md`	Retired
10.5	The 2D solver on Metal	`plans/2d/2D_GPU_METAL_STRATEGY.md`	Planned
10.6	A bed-step treatment at finite-volume junction faces	none — `ROADMAP.md` §1.1 only	Planned
10.7	The Vasconcelos and Wright (2009) hybrid flux for the TPA filling front — no longer required: the divergence it answered was removed on 2026-09-12	`plans/FV_SLOT_TPA_REVIEW_2026-08-31.md`, `plans/ISSUE_MIXED_FLOW_TPA_UF.md`	Retired
10.8	LID units as storage nodes, the hydraulic side	`plans/LID_StorageNode_Redesign.md`; `ROADMAP.md` §6	Planned
10.9	Retired formulations	`plans/IMEX_LOCAL_INERTIAL_IMPLEMENTATION_PLAN.md`; `ROADMAP.md` §7; `src/engine/input/handlers/OptionsHandler.cpp`	Retired

Table 10-1 Formulations recorded in this chapter

10.1 Chebyshev spectral geometry for irregular sections

The IRREGULAR, STREET and CUSTOM sections of Chapter 4 are evaluated from 51-row tables of area, width and hydraulic radius against depth, interpolated piecewise-linearly. The forward lookups are nearly free — about 15 ns — but the solvers spend their time in the inverse direction: Newton on \(A(S)\) costs 45 to 300 forward lookups, and the finite-volume depth inversion of §8.4.3 must use a derivative-free Brent search because the tabulated width is not the derivative of the tabulated area. On a transect-heavy dynamic-wave deck the irregular chain measured 56 % of the routing wall clock.

The first study (plans/CHEBYSHEV_XSECT_FEASIBILITY_2026-08-21.md) fitted global Chebyshev series to the table interpolant against the 1e-8 parity bar of the bit-exact tier and returned a no-go on replacement: the target has fifty corners, the error frontier stalled five orders short, the fits rang and were not monotone. It found a conditional go on a narrower use — a degree-12 inverse seed feeding one or two Newton polishes against the exact evaluators, 1.6–3.4× on the finite-volume inversion — inside the tolerance-gated fast tier, never the default.

The plan of record (plans/CHEBYSHEV_XSECT_GEOMETRY_PLAN.md, 2026-08-22) reframes the target and supersedes that verdict. The reference is the exact continuous geometry between stations — width piecewise linear, area piecewise quadratic — of which the table is itself a sampling with its own error (normalized: area \(10^{-3}\), width 0.3, radius 0.1), and the yardstick becomes "as accurate as the table", demonstrated end to end on deck outputs. Per transect, at resolve time: up to four panels with breaks placed first at detected discontinuities of the effective hydraulic radius (an interior station submerging switches the conveyance grouping) and then at the strongest width kinks; per panel a degree-10 Chebyshev fit of \(p \approx \sqrt{dA/dy}\), the area taken as the exact antiderivative of \(p^{2}\) — monotone by construction, pinned to \(A(y_{full})\) — the width as \(p^{2}\), the hydrostatic moment as the antiderivative of the area, the radius fitted per panel at degree 12, plus a global seed for \(y(\sqrt{a})\). On 43 real transects the area error matches the table's budget, the seeded inverse beats the table on 43 of 43 with unconditional Newton convergence, and the radius fit beats the table including the discontinuous transects. A per-transect acceptance check falls back to the table, so the feature is safe on geometry the study never saw. Selection is XSECT_GEOMETRY TABLES (default) or CHEBYSHEV, behind a CMake switch, with phases from a benchmark baseline through the fitter, the evaluators and parity harness, the finite-volume and dynamic-wave integrations and an end-to-end sign-off, and kill criteria that leave the fitter as a study artifact if the deck-level gates fail.

No part of this is in the tree: there is no XSECT_GEOMETRY option, no Chebyshev evaluator in src/, no OPENSWMM_CHEB_XSECT switch. Planned

10.2 An implicit finite-volume solver

plans/FV_IMPLICIT_HYPRE_SOLVER_EXPLORATION.md (2026-05-13) is an exploration, by its own status line, of a fully implicit finite-volume reformulation of dynamic-wave routing. Each conduit is cut into \(N_c \ge 1\) control volumes carrying wetted area, with discharge staggered at faces; junctions are zero-length volumes with a single head. Backward Euler (or BDF-2) with new-time face fluxes gives a nonlinear residual solved by Newton, its sparse linear systems preconditioned by hypre BoomerAMG under GMRES or CG, with an Eigen back end when hypre is absent. The motivation is the stiffness the Picard solver meets in looped surcharged systems and mixed flow, and the wish for steps set by accuracy rather than stability; the recommendation is a parallel, opt-in solver preceded by a one-trunk-main feasibility prototype.

The prototype was never built, and two premises have moved beneath the plan. The explicit solver of Chapter 8 took the FLOW_ROUTING FV slot and the FV_ prefix — its FV_CELL_LENGTH and FV_MIN_CELLS occupy the ground the plan's FV_CELLS_PER_CONDUIT and FV_TARGET_CFL were drawn for, and FV_DYNWAVE does not exist — and the hypre dependency the plan is named for left the build with the implicit 2D integrators (§10.9). As a plan of record it would need re-founding on the current tree. Planned

10.3 The 1D finite-volume solver on Kokkos backends

The explicit solver of Chapter 8 was planned with a device backend (plans/EXPLICIT_FV_KOKKOS_1D_SOLVER_PLAN.md §5): reconstruction, HLLC fluxes with the upwinded species flux, a per-chain tridiagonal dispersion solve, cell and node updates (the face-to-node gather as a deterministic two-pass CSR), a Courant reduction, wet–dry flags and a small host pass for structures, with state device-resident across every substep of a routing step in deliberate parallel with the 2D plugin, and a shared device cross-section module gated by a bit-exact extraction test.

What shipped is the option surface and the loader. FV_BACKEND accepts CPU, AUTO, OMP, CUDA, HIP and SYCL, FV_MIN_PARALLEL_CELLS gates AUTO, and NetworkSolverFactory searches the plugin path for openswmm_gpu_<backend> and requires two symbols — openswmm_gpu_probe and openswmm_make_gpu_network_solver — before accepting a plugin, falling through to the CPU solver otherwise. No plugin in the tree exports the network-solver symbol; the 2D plugins export the surface and inertial-marcher factories only, so every device request resolves to the CPU solver with a notice, as ROADMAP.md §7 records: "the same loader hooks and option surface but no plugin implementation yet".

The benchmark campaign that was to gate the port (§7B.3) returned a no-go at the sizes measured: 2000 conduits at four cells each does not reach the 20 000-cell gate, and at that size the solver runs 189× serial dynamic wave, so no parallel efficiency makes it competitive there. The plan is careful about scope — the dynamic-wave prototype of §10.4 measured a Picard iteration, the wrong shape for a device, whereas the explicit solver is local-stencil work of the right shape — and names the undecided regime, \(10^{5}\) to \(10^{6}\) cells, together with the cheap way to settle it: the solver's OpenMP thread scaling at those counts. The Kokkos implementation is on hold pending that measurement (owner ruling 2026-08-08). Planned

10.4 Dynamic wave on GPU

plans/1d/1D_DYNWAVE_GPU_KOKKOS_STRATEGY.md (2026-06-13) asked whether the link–node solver of Chapter 3 could be offloaded through Kokkos, and answered with a working prototype (plans/prototypes/dynwave_kokkos/) reproducing the solver's data flow — the structure-of-arrays conduit tile, a conveyance kernel, the node-inflow scatter as atomics, the Picard depth update and the global convergence reduction — with the same kernel bodies called from a serial reference and the Kokkos loops. The port is faithful: bit-exact on one thread, within \(1.5 \times 10^{-8}\) ft on four. The performance result set the direction aside: on the launch-cheap OpenMP backend the parallel path was slower than serial at every size to a million nodes (0.06× at 200, 0.81× at \(10^{6}\)), because three to twenty Picard sweeps per step, each with a global reduction and a host synchronization, are latency-bound until the element count is enormous; a discrete GPU's crossover lies further right, at sizes that do not occur in 1D practice.

The recommendation, adopted, is not to pursue single-model offload of the dynamic-wave solver. Two reframings survived — Kokkos as a portability layer for the already-threaded CPU loops, and ensemble parallelism (many models of one topology in one launch), which the plan identifies as the credible payoff and for which no design exists. ROADMAP.md §7 lists GPU solvers as no longer deferred on the strength of the 2D plugins and the hooks of §10.3; the dynamic-wave offload itself is retired on measurement. Retired

10.5 The 2D solver on Metal

The Kokkos backends of §9.11.1 have no Apple-GPU target, because Kokkos has no production Metal backend. plans/2d/2D_GPU_METAL_STRATEGY.md (2026-07-10) scopes the one path that closes the gap: a native Metal plugin, libopenswmm_gpu_metal.dylib, implementing the same ISurfaceSolver interface and C plugin ABI as the Kokkos plugins and loaded through the existing factory with three additive lines. The hard constraint is precision: Apple Silicon GPUs implement no double, so the plan proposes mixed precision by default (fluxes in float, the conserved accumulation and coupling ledger in higher precision), double-single emulation where validation demands it, and an acceptance bar of parity within a documented tolerance rather than bit-identity; unified memory makes the host–device hand-off a coherency matter rather than a copy.

The plan is stale in one structural respect: its port surface is the implicit CVODE solver — nine right-hand-side kernels, a Jacobi preconditioner and a matrix-free SPGMR loop, with a single-precision SUNDIALS linked into the plugin — and that solver was retired on 2026-07-29 (§10.9). The engine a plugin must now reproduce is the explicit marcher of §9.5, for which the ABI already reserves the inertial-solver factory the Kokkos plugins implement. Packaging, discovery, fallback and the precision analysis carry over; the kernel inventory, milestones and validation cases do not. Planned

10.6 A bed-step treatment at finite-volume junction faces

ROADMAP.md §1.1 records, under "Peak attenuation at the default mesh", that one cell per conduit attenuates the reference model's peaks by 37 % on average (7 % at \(\Delta x = 20\) ft), and that the cause is geometric rather than diffusive: a cell-centred scheme places a single cell's bed at the conduit mid-point, so every manhole presents an artificial bed step — §8.3 and §8.10 describe the mechanism and its measured cost. Higher-order reconstruction does not rescue it, because the error is in where the bed is, not in how the state varies across the cell. FV_CELL_LENGTH is the present workaround; the roadmap names "a bed-step treatment at junction faces" as the fix.

No plan file exists for this item. It is one line in the roadmap's remaining-scope list. What such a treatment must do follows from Chapter 8: the junction face of §8.6.1 would see, on each side, the conduit's true invert at the node rather than the cell's mid-point bed, so that the hydrostatic reconstruction of §8.5.1 is performed against the real step at the manhole and not a fictitious one — the reconstruction a resolved mesh's interior faces already perform, which is why refinement removes the attenuation. Whether by extrapolating the bed to the face, a junction-face reconstruction, or a two-sided invert per cell, is undecided. Planned

10.7 The TPA high-celerity filling divergence and the Vasconcelos and Wright (2009) hybrid flux

The two-component pressure approach of §8.4.5 shipped with one pinned defect: on filling and reflection decks at high acoustic celerity — the laboratory fixture at \(a = 150\) m/s — explicit Euler diverged at the reflected surge within seconds, a temporal odd–even pressure–vacuum oscillation inside the flagged region that a local conservative filter, tried and measured, could not damp. plans/ISSUE_MIXED_FLOW_TPA_UF.md records the hybrid flux of Vasconcelos and Wright (2009) — the paper's own post-shock treatment — as the documented contingency and an explicit non-goal; ROADMAP.md §1.1 names it the designated fix path, with the gate FvTpa.KnownIssueHighCelerityFillingDiverges pinning the divergence.

The review in plans/FV_SLOT_TPA_REVIEW_2026-08-31.md §5 then found a family of divergences attributed to the closure to be an artefact of the dynamic-wave step floor: the finite-volume substep was clamped up to the legacy 1 ms MIN_TIMESTEP, discarding the Courant bound on short cells at every FV_CFL, and the retry loop accepted the violating step silently. A \(10^{-6}\) s loop guard removed those, but the \(a = 150\) m/s Euler mode survived and the pin stood, correctly attributed. It fell on 2026-09-12 with the slot/free-surface wave-speed bound of the closure-kernel program: Davis's symmetric estimate had carried the acoustic celerity into the wave entering the free-surface side of a pressurization front, and bounding that wave by the Rankine–Hugoniot bore speed removes the mode. The fixture completes at 0.000 % continuity at \(a\) = 150, 300, 600, 1000 and 3000 m/s, and the gate is now the positive test FvTpa.HighCelerityFillingCompletes (tests/unit/engine/test_fv_tpa_closure.cpp). The roadmap entry, §8.4.5 and one test comment still cite the old gate name.

The hybrid flux is therefore no longer a required fix and has not been implemented. It remains what the issue called it — a contingency should a future deck reproduce the post-shock oscillation the paper describes — with no plan to build it. Planned

10.8 LID units as storage nodes, the hydraulic side

plans/LID_StorageNode_Redesign.md and ROADMAP.md §6 record the design for moving LID units out of the subcatchment and into the network, on the argument that a green-infrastructure train's interactions — head-dependent flow between units, backpressure from a saturated downstream unit, control structures between units, an underdrain that sees the network's grade line — are hydraulic phenomena a one-way runon cascade cannot mediate. The hydraulic side uses machinery this manual already documents. A storage node (Chapter 5) gains a [LID_LAYERS] attribute giving each layer a fractional depth, porosity, conductivity and moisture limits; its storage curve is corrected for the void fraction,

\[V(d) = A\left\lbrack h_{surface} + \varphi_{media}\,h_{media} + \varphi_{gravel}\,h_{gravel} \right\rbrack\]

so the head the node reports is the water surface accounting for media occupancy, and that head is what the connected links see. The underdrain is an ordinary orifice at the gravel-base offset — its discharge follows the orifice law of Chapter 6 on the head difference to the receiving node and reverses when the network surcharges above it, with no special casing; surface overflow is a weir at the surface elevation; mid-media outlets are further offset orifices; a chain of units is ordinary node–link topology with control rules on the links between them. Contributing areas drain to the node through the existing runon mechanism. What is new — a kinematic Richards system for the unsaturated media, its moisture-to-head mapping and a flux limiter — is hydrology.

The status is "design recorded, no implementation". The plan's code sketches target the legacy C sources (objects.h, lid.c, node.c) rather than the current C++ engine, so an implementation would begin by re-siting them. Planned

10.9 Retired formulations

The IMEX local-inertial integrator. Retired plans/IMEX_LOCAL_INERTIAL_IMPLEMENTATION_PLAN.md proposed advancing the 2D surface with SUNDIALS ARKStep as an implicit–explicit integrator: a diffusive-wave Phase 1 with the lateral diffusion implicit, a local-inertial Phase 2 with the state enlarged to cell volumes and edge discharges, and a Froude-gated blend selected by MOMENTUM and FROUDE_BLEND, preconditioned by Casulli's elliptic free-surface operator under hypre BoomerAMG. Phase 1 was implemented as ArkodeSurfaceSolver and measured on 2026-06-26: correct and conservative, and structurally slower than CVODE — 2.7× on the road-culvert case, 4.9× to 31× on a synthetic basin, worsening with scale, at twenty times the Newton and fifty-nine times the Krylov work — because in the diffusive wave the only stiff term is the one the split made implicit, so the split merely exchanged an adaptive-order BDF for a fixed-order DIRK on the same global solve. The plan's header records the premise as refuted; Phases 2 and 3 were never built, and the integrator went with the CVODE stack.

The CVODE and ARKODE integrators. Retired The implicit integrators were retired on 2026-07-29: the explicit marcher of §9.5 outperformed them on the benchmark models and removed the SUNDIALS and hypre dependencies (ROADMAP.md §7). INTEGRATOR accepts EXPLICIT only; the fourteen keys that configured the retired stack, and any other INTEGRATOR value, are accepted with WARNING 104 and ignored on file load so legacy models still open, and are errors on the programmatic option-set path (§9.11.2).

ADVECTION YES. Retired The local-inertial law with a staggered upwind convective difference added to (9-7) on wet–wet faces (Stelling and Duinmeijer, 2003) was the first attempt to supply the term §9.2.1 drops. It was superseded on 2026-09-06 by MOMENTUM_EQUATION FULL_SWE, which supplies the term conservatively with shock capturing. The key is deprecated rather than removed: a deck that carries it is warned and keeps the physics it asked for (§9.2.3, Table 9-2).

VIRTUAL_JUNCTION_MOMENTUM FULL. Retired The cross-junction momentum term FULL added at a virtual junction under dynamic-wave routing was measured defective, not merely inaccurate: it is sign-inverted with respect to the per-link convective term it supplements (at constant discharge \(\Delta(v^{2}A) = -v^{2}\Delta A\)) and applied to both adjacent links on top of each link's own term. On the SWASHES MacDonald periodic case it destroyed 224–325 % of the routed volume; negating it restored conservation but left 5.24 % \(L^{1}\) error against 0.163 % for BASIC, so no term-level correction was worth keeping. Retired 2026-08-14: FULL is accepted, warned and treated as BASIC (src/engine/input/handlers/OptionsHandler.cpp). Under finite-volume routing the question does not arise — a virtual junction is one interior face, and conservation across it is a property of the scheme (§8.6.2).

The FV_NODE_* keys. Retired Five keys are retired in two tiers, on purpose. FV_NODE_COUPLING, FV_NODE_DT and FV_NODE_PICARD (2026-08-29) once selected explicit versus semi-implicit storage-node coupling, whether the node bound (8-32) was armed, and the number of correction sweeps; storage nodes are now always semi-implicit, the bound always armed, the correction one sweep — each the former default. A deck spelling the former default is not warned; one asking for the retired behaviour (EXPLICIT, NONE, sweeps above one) is warned at open, since a calibrated model would otherwise change answers silently. FV_NODE_CELL_COUPLING and FV_JUNCTION_MODEL were already no-ops — the bucket-junction model and the coupled-star correction were superseded by the pass-through interface of §8.6.1 — and stay silent (src/engine/input/handlers/OptionsHandler.cpp; §8.9).

The works cited above are listed in §8.12 (Vasconcelos, Wright and Roe, 2009) and §9.13 (Stelling and Duinmeijer, 2003).

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