Advection schemes

The runtime ships four flux-form advection schemes, each behind the abstract type AbstractAdvectionScheme declared in src/Operators/Advection/schemes.jl. Upwind, slopes, and standard PPM use directional sweeps. Lin–Rood couples the two horizontal directions; configured runs default its vertical sweep to FV3's kord = 8 tracer profile, while the low-level Julia constructor retains upwind. On the cubed sphere, PPM can also select that profile with vertical = "fv3_kord8". The model-facing operator interface dispatches to the appropriate transport path.

Scheme	Smooth-flow accuracy	Monotone?	Positive?	LL	CS	RG
UpwindScheme	1st order (donor cell)	yes (trivially)	preserves a non-negative input under its CFL contract; also supports signed tracers	yes	yes	yes — RG's only option today
SlopesScheme{L}	2nd order in smooth regions (van Leer / Russell-Lerner)	yes if L = MonotoneLimiter (default)	no zero clamp in the default signed path; PositivityLimiter is explicit opt-in	yes	yes	no (the face-indexed Strang path restricts to AbstractConstantScheme)
PPMScheme{L,V}	2nd order: limited Colella–Woodward (1984) edges set a Russell–Lerner slope. With L = CW84Limiter the flux integrates the limited parabola (the complete CW84 scheme). With V = FV3ScalarProfile the vertical sweep integrates FV3's kord = 8 parabola (3rd order or better in smooth regions)	profile-limited with MonotoneLimiter, but the 1-D update over- and undershoots; with CW84Limiter monotone in uniform 1-D flow (Courant number ≤ 1)	signed; small negative column means observed in CS runs with MonotoneLimiter. CW84Limiter keeps a non-negative tracer non-negative in every sweep in which no cell exports more than its air mass. The FV3 vertical profile is positive definite	yes (default vertical only)	yes — covered by test/core/test_cubed_sphere_advection.jl and test/core/test_fv3_vertical_profile.jl	no (same rejection)
LinRoodPPMScheme{ORD}	piecewise-parabolic; ORD ∈ {5, 7} selects the boundary stencil	profile-limited, but the full split update can undershoot	signed; not positivity-preserving	n/a	yes — uses FV3 cross-term advection (fv_tp_2d_cs!)	n/a

The accuracy column describes the per-face reconstruction in smooth regions; near limiters / discontinuities high-order reconstruction can drop to first order locally. LinRoodPPMScheme's ORD ∈ {5, 7} selects the boundary stencil for cross-panel faces; the in-panel reconstruction is the same FV3 PPM either way. ORD=7 is not a globally seventh-order transport method. Spatial reconstruction order also does not establish the temporal order of the complete split update.

Selecting a scheme

The TOML runner defaults to scheme = "upwind" when the selector is omitted. Choose the algorithm explicitly when comparing runs:

[advection] settings	Runtime scheme	Vertical transport
scheme = "upwind"	UpwindScheme()	upwind
scheme = "slopes"	SlopesScheme(MonotoneLimiter())	slopes
scheme = "ppm"	PPMScheme(MonotoneLimiter())	PPM-informed slopes, as horizontally
scheme = "ppm", limiter = "cw84"	PPMScheme(CW84Limiter())	complete CW84 PPM, as horizontally
scheme = "ppm", vertical = "fv3_kord8"	PPMScheme(MonotoneLimiter(), FV3ScalarProfile()) (CS only)	FV3 scalar_profile, kord = 8, positive definite
scheme = "ppm", vertical = "fv3_kord8_signed"	PPMScheme(MonotoneLimiter(), FV3ScalarProfile(; positive_definite = false)) (CS only)	FV3 scalar_profile, kord = 8, for signed tracers
scheme = "linrood", ppm_order = 5	LinRoodPPMScheme(5; vertical = PPMScheme(; vertical = FV3ScalarProfile())) (CS only)	FV3 scalar_profile, kord = 8 (default)
scheme = "linrood", ppm_order = 7	LinRoodPPMScheme(7; vertical = PPMScheme(; vertical = FV3ScalarProfile())) (CS only)	FV3 scalar_profile, kord = 8 (default)
scheme = "linrood", vertical = "upwind"	LinRoodPPMScheme(5; vertical = UpwindScheme()) (CS only)	upwind (explicit legacy choice)
scheme = "linrood", ppm_order = 7, vertical = "fv3_kord8"	LinRoodPPMScheme(7; vertical = PPMScheme(; vertical = FV3ScalarProfile())) (CS only)	FV3 scalar_profile, kord = 8

Omitting ppm_order for Lin–Rood selects 5, and omitting vertical selects the GCHP-matched fv3_kord8 profile, in forward runs and in the inversion driver alike: every Lin–Rood vertical choice has an adjoint (see the FV3 profile's adjoint). The Julia constructors LinRoodAdvectionSpec(order) and LinRoodPPMScheme() remain upwind. Setting ppm_order with scheme = "ppm" is an error: standard split PPM has no order selector. vertical is only accepted with scheme = "ppm" or "linrood", and limiter ("monotone", the default, or "cw84") only with scheme = "ppm". NoLimiter and PositivityLimiter are selected through the Julia constructors. Packed tracer arrays, GPU workgroup sizes, and copy-back or ping-pong execution are implementation choices within a scheme, not additional algorithms.

Diagnosed slopes with the Russell–Lerner flux (SlopesScheme)

Per-cell linear reconstruction with a monotone limiter by default:

$${\chi }_c(x)\approx {\chi }_c+s_c(x-x_c)$$

where s_c is the slope estimate. The face flux through the right-of-cell-c face uses the Courant-fraction formula (Russell & Lerner, 1981):

$$F_{c+1/2}^{n+1/2}=F_{c+1/2}^n({\chi }_c^n+s_c^n\cdot (1-{\alpha }_{c+1/2})/2)$$

with α = F / m the local Courant fraction (F is the per-substep mass amount stored in the binary, so the Δt factor is already absorbed into F — see Mass conservation for the flux_kind = :substep_mass_amount convention). The slope s_c is re-diagnosed from the neighbouring cell means at every sweep (van Leer 1977; MonotoneLimiter takes the minmod of the centred and twice the one-sided differences), so the scheme carries no slope state between sweeps. The production kernel is _slopes_face_flux in src/Operators/Advection/reconstruction.jl, with the canonical formula derivation in that function's docstring.

Properties:

• 2nd order in smooth regions even with NoLimiter (the centered slope itself is a 2nd-order reconstruction; what MonotoneLimiter buys you is monotonicity at discontinuities, at the cost of dropping locally to 1st order at limiter saturation).

• Monotonic with MonotoneLimiter (van Leer minmod); does not produce new extrema across faces.

• Valid for signed mixing ratios and equivariant under a constant VMR offset. The monotone profile is limited relative to neighbouring values, never relative to tracer zero.

• Mass-conservative to floating-point via the flux-form telescoping argument (see Mass conservation for the precise statement and round-off bounds).

Relation to TM5's slopes advection

SlopesScheme is not TM5's slopes scheme, although it uses the same face flux. TM5 (dynamu, dynamv, dynamw in advectx.F90, advecty.F90, advectz.F90) carries the first moments rxm, rym, rzm of every tracer as prognostic state. Each sweep transports them with their own flux and update equation, and the moments are also updated by TM5's convection and emissions. TM5 limits a moment only for positivity, |rxm| ≤ rm, and does not use a minmod limiter.

test/core/test_slopes_tm5_reference.jl checks the relation against an independent transcription of TM5's dynamu on a periodic row (test/helpers/tm5_dynamu_reference.jl):

• With rxm re-diagnosed by the minmod rule at the start of every sweep, the TM5 update reproduces SlopesScheme() to rounding, and with rxm := 0 it reproduces UpwindScheme(). The flux formula is shared; the moment equation is what is missing.

• With the prognostic moment, TM5 is much less diffusive. A Gaussian of half-width 6 cells on a 120-cell ring keeps 0.952 of its peak after 10 revolutions at Courant number 0.5 in TM5, against 0.719 with SlopesScheme (0.830 with PPMScheme). After 10 revolutions the SlopesScheme L2 error is 4.6 to 6.8 times TM5's at Courant numbers 0.05 to 0.75.

• TM5's positivity clamp refers to zero, so TM5 is not equivariant under a constant offset (transporting c − 1/2 is not transporting c and subtracting 1/2). SlopesScheme limits against neighbouring values and is offset-equivariant, which signed anomaly tracers require.

Known limitation: no prognostic-moment (TM5) slopes scheme

No advection option reproduces TM5's slopes advection. A prognostic-moment scheme is postponed: it would need moment state for every tracer carried through emissions, convection and diffusion, moment rotation at cubed-sphere panel seams, an adjoint, and a limiter that is safe for signed tracers. Runs made with scheme = "slopes" are diagnosed-slope (van Leer) results, not TM5-equivalent ones.

PPM-informed slopes (PPMScheme)

PPMScheme builds the piecewise parabola of Colella & Woodward (1984) in each cell,

$${\chi }_c(\xi )={\chi }_c^L+\xi (\Delta {\chi }_c+{\chi }_c^{(6)}(1-\xi )),0\le \xi \le 1,$$

with Δχ_c = χ^R_c − χ^L_c and χ^{(6)}_c = 6(χ_c − ½(χ^L_c + χ^R_c)). The edge values come from the uniform-index fourth-order formula χ_{c+½} = 7/12 (χ_c + χ_{c+1}) − 1/12 (χ_{c−1} + χ_{c+2}) and are limited with the Colella–Woodward monotonicity conditions (_ppm_limit_profile).

The face flux does not integrate the parabola. Only the limited edge on the outflow side is used: it sets the edge offset s_x = m_c (χ^R_c − χ_c) (outflow to the right) of the Russell–Lerner flux above (_slopes_face_flux). The update is therefore second order, with a PPM-informed slope. Donor layers within two layers of the model top or surface have no slope (upwind flux).

This default is not monotone: the unlimited fourth-order edge can lie outside the range of its neighbouring cell means, and the linear flux ignores the parabola's curvature. A box or spike advected 100 steps around a 40-cell ring undershoots to −0.16 (box, Courant number 0.7) and overshoots to 1.015 (box, 0.3) on a [0, 1] field (test/core/test_ppm_cw84_limiter.jl).

Complete CW84 PPM (limiter = "cw84", PPMScheme(CW84Limiter())). Three changes give the scheme of Colella & Woodward (1984):

1. Edges use van Leer-limited slopes (CW84 eqs. 1.7–1.8),

$${\chi }_{c+1/2}=\frac{1}{2}({\chi }_c+{\chi }_{c+1})-\frac{1}{6}(\delta {\chi }_{c+1}-\delta {\chi }_c),\delta {\chi }_c=\mathrm{minmod}(\frac{1}{2}({\chi }_{c+1}-{\chi }_{c-1}),2({\chi }_{c+1}-{\chi }_c),2({\chi }_c-{\chi }_{c-1})),$$

so each edge lies between its two neighbouring cell means. Where neither slope is limited this equals the fourth-order edge above.

2. The profile is limited with the same monotonicity conditions.

3. The flux is the mass flux times the parabola's mean over the swept fraction α = |F|/m of the donor cell (CW84 eq. 1.12). For outflow to the right this is χ_c + (1 − α)(b_R − α b_0) with b_R = χ^R_c − χ_c and the curvature b_0 = (χ^L_c − χ_c) + (χ^R_c − χ_c); it enters _slopes_face_flux as the moment s_x = m_c (b_R − α b_0) (_ppm_raw_moments).

The stencil is the same six cells, so halo widths do not change. In uniform one-dimensional flow with Courant numbers up to 1 the update is monotone; on the ring above it stays in [0, 1] in Float32 and Float64. Every limited parabola is bounded by the neighbouring cell means, so within one sweep a cell's tracer mass stays non-negative as long as the fractions of the cell swept out through its two faces sum to at most 1. When the runtime chooses the subcycle count, its palindrome budget guarantees this for every pass: each pass removes at most cfl_limit / 2 of the mass the cell holds when the pass starts (see "CFL handling and subcycling" below). Cubed-sphere binaries with the binary_schedule substep contract run n = 1 and rely on the write-time gate, which checks the same budget against the window's masses rather than each step's, so there positivity is measured, not proven. Advecting one period of a sine at Courant number 0.4, the L2 error falls by 4.05 from 80 to 160 cells (3.46 for the default) and is 18% lower at 160 cells; at 20 cells it is 29% higher, because the profile limiter flattens the sine's extrema. The cubed-sphere adjoint supports both the monotone and CW84 limiters (_ppm_face_coeffs); the run log labels the option PPM, CW84.

Vertical reconstruction. On the cubed sphere, PPMScheme(; vertical = FV3ScalarProfile()) (TOML vertical = "fv3_kord8") replaces the vertical face flux with the profile GEOS-Chem High Performance uses for tracers: FV3's scalar_profile with kord = 8. Its edges solve a compact tridiagonal system weighted by layer air mass. They are limited with large-scale constraints, Huynh's second constraint and a positive-definite limiter. The flux integrates the donor layer's parabola over the swept fraction. The positive-definite profile flattens layers whose mean is not positive, so tracers that become negative (flux anomalies, for example) need vertical = "fv3_kord8_signed" (FV3's iv = 1). Equations, limiter details and tests are in Vertical transport, section 4.

Measured accuracy. A Gaussian bump is translated 30 layers on a uniform grid; test/core/test_fv3_vertical_profile.jl holds a version of this as a test. Doubling the bump width from 6 to 12 to 24 layers cuts the RMS error:

• 4× for slopes and the default PPM (second order);

• 11–23× for the FV3 vertical profile (third order or better);

• 2× for upwind.

The FV3 profile is 4–900 times more accurate than the default for widths of 3–24 layers and Courant numbers of 0.01–0.3 (output: /temp1/cfranken/scratch/fv3_vertical/translation_probe.txt).

Properties:

• Second order in smooth regions (horizontal sweeps and default vertical); local accuracy drops to first order at limiter saturation.

• Monotonicity controlled by the limiter parameter; without a limiter PPM is not monotone and can produce small oscillations near discontinuities. The FV3 vertical profile is positive definite but not monotone.

• Shipped on lat-lon AND cubed-sphere structured layouts (the CS case is exercised by test/core/test_cubed_sphere_advection.jl). The FV3 vertical profile is cubed-sphere only. Face-connected PPM for the reduced-Gaussian topology is not currently wired — and neither is SlopesScheme on RG. The face-indexed Strang path restricts to AbstractConstantScheme, so UpwindScheme is the only advection option for RG production runs today.

For cubed-sphere runs that need the FV3 cross-term advection at panel edges and permit signed undershoots, LinRoodPPMScheme (next section) is the relevant variant. Default monotone PPM limits the reconstructed profile, but the full CS update has produced small negative column means in real-input tests. Use upwind under its CFL contract when non-negativity is required, or validate the higher-order scheme's bounds on the intended workload.

Lin-Rood PPM with cross-term (LinRoodPPMScheme{ORD})

The cubed-sphere variant. Extends PPM with the two-step Lin-Rood splitting (fv_tp_2d_cs! in src/Operators/Advection/linrood_horizontal.jl) so the X and Y sweeps see each other's intermediate fluxes via the inner-edge flux-and-slope rotation that FV3 uses internally. The runtime pairs this horizontal update with vertical upwind. Two edge-value families are selectable:

ORD	Interior reconstruction	Panel-edge treatment
5	Colella–Woodward fourth-order edges with extremum flattening	none: the fourth-order edges read the neighbor panel's halo cells
7	Same as ORD=5	FV3's cube-edge values (below)

With ORD = 7 every face carries one edge value, used by both cells beside it, as FV3's al(i) in tp_core.F90 (xppm/yppm, iord < 8). With face f between cells f−1 and f and the panel edges at faces 1 and Nc + 1:

$$\begin{array}{rlrl}{q}_1& =\frac{1}{2}[(\frac{3}{2}{q}_0-\frac{1}{2}{q}_{-1})+(\frac{3}{2}{q}_1-\frac{1}{2}{q}_2)]& & \text{panel edge (Putman \& Lin 2007, App. C, eq. 47)}\\ q_2& =\frac{5}{14}{q}_1+\frac{11}{14}{q}_2-\frac{2}{14}{q}_3,q_0=-\frac{2}{14}{q}_{-2}+\frac{11}{14}{q}_{-1}+\frac{5}{14}{q}_0& & \text{one-sided, beside the edge}\\ q_f& =\frac{7}{12}(q_{f-1}+q_f)-\frac{1}{12}(q_{f-2}+q_{f+1})& & \text{elsewhere,}\end{array}$$

and mirrored at Nc, Nc + 1, Nc + 2 (_ppm_edge_values_ord7). No edge of a cell beside a panel edge is built from a stencil that crosses it, so cell 1 has one parabola: the seam value on its left, the one-sided value on its right. FV3 weights the two seam extrapolations by the cell widths dxa; they are equal on both sides of an equiangular panel edge, and the code uses equal weights. Up to version 0.5.0 the seam value replaced only the edges at faces 1 and Nc + 1, so the face kernels rebuilt cell 1 with a cross-seam fourth-order left edge when they evaluated face 2 (issue #62). In solid-body rotation of a Gaussian blob about a tilted axis (one revolution in 16 Nc steps), the consistent edges lower the global relative L2 error of ORD=7 by 10–31% at C8–C64 (both panel conventions); in the two cell rows beside each panel edge the error changes by −9% to +17%. The forward and adjoint face kernels evaluate the same edge function (the adjoint on dual numbers), so the adjoint stays exact.

PPMScheme is the strict-structured PPM; LinRoodPPMScheme is the cross-term-aware CS-native variant. Lin-Rood no longer runs the zero-referenced fillz repair, because that repair destroys negative anomaly mass. The tradeoff is explicit: a non-negative input can develop small negative undershoots even while carrier air mass remains positive and global tracer storage is conserved. Choose it for signed/anomaly transport or after validating that this boundedness tradeoff is acceptable for the tracer being simulated.

A divergence-damping (del-2) operator is layered on top (_divergence_damping_cs_kernel! in LinRood.jl) to suppress the small numerical noise that survives at the panel boundaries.

Limiters

Four selectable limiter types are declared in schemes.jl; their formulas live in src/Operators/Advection/limiters.jl and, for the PPM edges and moments, reconstruction.jl:

Limiter	What it enforces	Use case
NoLimiter()	unlimited centered slope / parabola	smooth-flow benchmarks where you want the order-N error rate without limiter clipping
MonotoneLimiter() (default)	van Leer minmod slope: minmod(central, 2forward, 2backward), where central = (forward + backward)/2. Bounds the reconstructed profile relative to neighboring values and supports signed tracers.	production runs, including anomaly tracers
CW84Limiter() (PPMScheme only)	the complete Colella–Woodward PPM: van Leer-limited edges, the monotone profile limiter, and the swept-parabola flux. Monotone in uniform 1-D flow; non-negative in every sweep in which no cell exports more than its air mass.	tracers that must stay non-negative under PPM (fossil and other emission tracers)
PositivityLimiter()	keeps the sign of the centered slope and clips its magnitude to |s| ≤ 2c, so both face values c ± s/2 stay non-negative (the moment form is the clamp |s_x| ≤ r_m). Rising and falling profiles are treated alike. Weaker than MonotoneLimiter: positivity-only, may still create new local maxima from large gradients. Requires a non-negative tracer.	tracers that must stay non-negative (mole fractions, water vapor, aerosol concentrations) AND tolerate occasional new maxima

Limiter primitives are written branchless (ifelse(a*b > 0, ..., 0)) in limiters.jl so they don't trigger warp divergence on the GPU.

The default monotone path deliberately has no post-step tracer clamp. Bounds on a local reconstructed profile do not establish positivity of the complete multidimensional CS update; monitor transported-field minima separately from mass totals. PositivityLimiter is a separate, explicitly zero-referenced policy and must not be used for anomaly tracers.

CFL handling and subcycling

The binary's stored fluxes are per-substep mass amounts (flux_kind = :substep_mass_amount), already pre-divided by the substep count for the active window. The Courant number is therefore a pure mass ratio — no Δt factor:

$${\alpha }_{c+1/2}=\frac{|F_{c+1/2}^n|}{m_c^n}$$

One transport step applies each direction twice (X Y Z | midpoint | Z Y X, H V | midpoint | V H on reduced Gaussian), so the runtime budgets one palindrome's worth of outflow per cell against the smaller of the cell's masses at the start and at the end of the step,

$$r_c=\frac{2({\mathrm{out}}_x+{\mathrm{out}}_y+{\mathrm{out}}_z)}{min(m_c,m_c^{\mathrm{end}})},{\mathrm{out}}_d=max(0,-F_{d,\mathrm{lo}})+max(0,F_{d,\mathrm{hi}}),m_c^{\mathrm{end}}=m_c+2\sum _d(F_{d,\mathrm{lo}}-F_{d,\mathrm{hi}}),$$

and when max r_c exceeds cfl_limit it subcycles the whole step:

n = ceil(max_c r_c / cfl_limit)

and runs the symmetric composition

$$S(\Delta t)=[X_h{Y}_h{Z}_h{]}^n\circ V(\Delta t)\circ [Z_h{Y}_h{X}_h{]}^n,h=\frac{\Delta t}{2n},$$

where X_h is an x sweep with every flux scaled by 1/n and operator products act right to left: n forward triples X → Y → Z, the midpoint operators V once, and n reverse triples Z → Y → X. With ψ(h) = Z_h Y_h X_h the forward half is ψ^n and the reverse half (ψ^*)^n, where ψ^*(h) = X_h Y_h Z_h = ψ(−h)^{-1} is its adjoint, so S(−Δt) = S(Δt)^{-1}: the composition is symmetric, hence second order like the palindrome itself, with the same 6n sweeps; at n = 1 it is the plain palindrome. FV3's fv_tracer2d subcycles the same way (its whole 2-D operator nsplt times at flux / nsplt).

What the budget guarantees. The midpoint operators do not change air mass, and each block of passes (one X_h Y_h Z_h or Z_h Y_h X_h) changes a cell's mass by the same amount, Σ_d (F_{d,lo} − F_{d,hi}) / n. The masses between blocks therefore lie on the straight line from m_c to m_c^end, so all of them are at least min(m_c, m_c^end). Within one block the passes remove at most (out_x + out_y + out_z) / n ≤ (cfl_limit / 2) min(m_c, m_c^end). Hence:

• every pass removes at most cfl_limit / 2 of the mass the cell holds when that pass starts, and

• the air mass stays at least (1 − cfl_limit / 2) min(m_c, m_c^end) > 0 throughout the step.

A reference mass of m_c alone, as the count used before this budget, bounds only the first block, because a cell that drains within the step starts its later blocks well below m_c. With one cell of mass 1 among cells of mass 100, inflow 1.51 in x and outflow 2.0 in y (m_c^end = 0.02), that rule chose n = 5, the reverse half drove the air mass negative, and PPM produced a mixing ratio of −1.7 × 10^43. The reference mass min(m_c, m_c^end) is the one of the write-time positivity gate (min(m, m_next), with the step's own end mass for m_next); here it gives n = 211. A cell with outflow whose reference mass is not positive has no valid count (the flux set would leave it with no or negative mass), and the count throws an ArgumentError. The budget is sufficient, not tight: it assumes that a block's passes remove all of their outflow before any of their inflow arrives. test/core/test_subcycle_composition.jl checks it on the draining cell above (x and y swapped too), solid-body rotation and strain on the cubed sphere, strain on lat-lon and an overturning circulation on reduced Gaussian. Up to version 0.5.0 each direction was subcycled on its own, X^n Y^n Z^n | Z^n Y^n X^n: the n passes of one leg then remove the full-flux outflow of that direction before any other direction refills the cell, so subcycling bounded the Courant number of a pass but not the mass loss within a leg, and exactly non-divergent flows drove air mass negative (issue #61).

_palindrome_subcycle_count and the per-cell _palindrome_outflow_ratio in src/Operators/Advection/cs_subcycling.jl are shared by all split-sweep topologies; _cs_static_palindrome_subcycle_count (same file), _structured_palindrome_subcycle_count and _face_indexed_palindrome_subcycle_count (subcycling.jl) evaluate them on each flux layout. See Operators on top of the binary for the write-time gate and the binary schedule, which sets n = 1 on cubed-sphere binaries with the binary_schedule substep contract.

All split-sweep topologies share one default cfl_limit, DEFAULT_CFL_LIMIT = 0.95, so when the runtime chooses the count no pass removes more than 47.5% of a cell's current mass and every cell keeps at least 52.5% of min(m_c, m_c^end):

Topology	Default cfl_limit	Notes
Lat-lon (structured-grid Strang, strang_split!)	0.95	Palindrome budget (was a per-direction 1.0 up to 0.5.0).
Reduced Gaussian (face-indexed Strang)	0.95	Palindrome budget over H and V; only UpwindScheme reachable today.
Cubed-sphere (SlopesScheme / PPMScheme, strang_split_cs!)	0.95	Palindrome budget.
Cubed-sphere (LinRoodPPMScheme)	not used	The LinRood path relies on the binary's adaptive substep schedule. Its in-kernel 0.9 × cell-mass clipping limits carrier outflow; it does not guarantee tracer positivity.

If you're running LinRoodPPMScheme on a flow with locally large Courant numbers, the recourse is to halve dt in the run config rather than rely on the operator to subcycle internally.

Strang palindrome and temporal accuracy

The full transport step is the time-symmetric composition

S(Δt) = X(Δt/2) Y(Δt/2) Z(Δt/2) ∘ V(Δt) ∘ Z(Δt/2) Y(Δt/2) X(Δt/2)

(with V(Δt) → V(Δt/2) S(Δt) V(Δt/2) when surface flux is on). Strang composition gives second-order splitting accuracy for sufficiently accurate subflows. The palindrome alone does not establish second-order accuracy of the implemented transport: reconstruction, halo evolution, and each subflow's time integration must also be validated. A cubed-sphere seam fixture shows first-order timestep refinement for both the original split PPM and its paired seam update. Conservation and temporal accuracy are separate requirements.

Convection and chemistry are NOT inside the palindrome — they are applied once per met window, post-palindrome — because their natural cadence is the met window rather than the advection sub-step, and they do not commute with advection at the per-substep level.

Panel-edge halo treatment (cubed sphere)

PanelConnectivity (src/Grids/PanelConnectivity.jl) carries a table of panel-edge mappings: for each panel p and each edge e ∈ {west, east, south, north}, what is the neighbour panel and what's the edge orientation (0 = aligned, 2 = reversed)? The default connectivity methods encode the GEOS-FP / GEOS-IT 6-panel arrangement and its gnomonic alternative.

Halo exchange supplies neighboring tracer and air-mass values for the reconstruction stencil. Matching air-mass fluxes alone does not guarantee matching tracer fluxes: both cells sharing a face must use the same tracer exchange at the same update stage.

Lin–Rood now shares the mean of the two panels' inner and outer face mixing ratios before applying either panel's final horizontal divergence. Only panel boundary faces are changed. Tangential indices follow the connectivity map; mixing ratios are scalars, and the mirrored air-mass flux supplies the normal sign. This makes the two tracer transfers cancel to roundoff. The adjoint applies the transpose of the same averaging operation.

The dimensionally split cubed-sphere Upwind, Slopes, and PPM paths assign each physical seam to the lower-numbered panel and that panel edge's local axis. Each X/Y group updates its panel-interior faces and all seams assigned to it. A seam transfer is reconstructed once, before any panel changes, then applied with opposite signs to both neighboring cells in the same group. This includes rotated contacts whose neighbor edge lies on the other local axis. The local panel kernels mask boundary fluxes so each physical exchange occurs once. Air and signed tracer mass therefore cancel across contacts; mirrored input mass fluxes remain required for consistency with the binary's continuity budget.

The forward tape records these same grouped sweeps. At fixed meteorology, the adjoint first collects both neighbors' output seeds, reverses interior sweeps, and then differentiates the shared seam reconstruction. This preserves the tracer-gradient contract without changing the tape format. The forward cache uses 12 × Nc × Nz × (Nt + 1) values, including air mass: 9.41 MB for C90 L66, 32 tracers, Float32. It has no six-tracer cap; Lin–Rood does not allocate this split-sweep buffer.

In a six-tracer C90 L66, 24-hour V100 test with TM5 convection and Dkg diffusion, sharing Lin–Rood seam estimates reduces maximum final relative tracer-total drift from 3.77e-5 to 6.98e-7 in Float32 and from 3.80e-5 to 7.93e-16 in Float64. The paired split PPM update reduces the corresponding maxima from 2.24e-5 to 8.17e-7 in Float32 and from 2.28e-5 to 9.91e-16 in Float64, with no global normalization. These are measured results for one forcing archive, not universal error bounds. Positive initial layers still develop negative column means: about -2.09e-10 mol/mol in Lin–Rood before and after its fix, and -4.27e-11 mol/mol in the corrected split PPM run. Conserving totals does not establish positivity or reference-model agreement. These seam measurements predate the subsequent conservative Dkg diffusion change, which further reduces the PPM Float32 maximum to 3.14e-7 on the same six-tracer workload.

An independent tilted solid-body rotation of a smooth Gaussian tracer on C8, C16, and C32 grids, with both panel conventions, gives slightly smaller area-weighted field errors for paired split PPM at quarter and full rotations. Float64 mass drift remains below 6.1e-15. This checks transported fields as well as totals; it is not a general temporal-order or positivity guarantee.

Reduced-Gaussian per-ring face segmentation

Adjacent rings on a reduced-Gaussian grid have different cell counts (nlon[j+1] ≠ nlon[j]), so the boundary between them cannot be covered by a one-to-one face mapping. _boundary_counts(nlon_per_ring) in src/Grids/ReducedGaussianMesh.jl segments each ring-pair boundary into lcm(nlon[j], nlon[j+1]) mini-faces — every cell on ring j contributes lcm/nlon[j] mini-faces, and every cell on ring j+1 receives lcm/nlon[j+1] mini-faces. Mass-flux pairing is then exact and the telescoping argument holds across the whole grid.

The cost is a higher face count than the cell count would suggest: adjacent rings with nlon = 108, 112 produce lcm = 3024 mini-faces between them. Performance-tuning notes live beside the implementation.

Where the schemes meet the code

Concept	File / function
Scheme abstract root	src/Operators/Advection/schemes.jl::AbstractAdvectionScheme
UpwindScheme, SlopesScheme{L}, PPMScheme{L,V}, LinRoodPPMScheme{ORD,Z}	src/Operators/Advection/schemes.jl
FV3 vertical profile (FV3ScalarProfile)	src/Operators/Advection/vertical_fv3_profile.jl
FV3 vertical-profile adjoint (frozen-limiter transpose)	src/Operators/Advection/vertical_fv3_profile_adjoint.jl::_sweep_z_panels_fv3_adjoint!
Limiter primitives (branchless, GPU-safe)	src/Operators/Advection/limiters.jl
Slopes face flux (Russell-Lerner formula)	src/Operators/Advection/reconstruction.jl::_slopes_face_flux
Structured-grid Strang palindrome	src/Operators/Advection/StrangSplitting.jl::strang_split!
Cubed-sphere Strang palindrome	src/Operators/Advection/CubedSphereStrang.jl::strang_split_cs!
CFL subcycle counts (lat-lon, reduced Gaussian)	subcycling.jl::_structured_palindrome_subcycle_count, _face_indexed_palindrome_subcycle_count
CFL subcycle count (CS) and the shared palindrome budget	cs_subcycling.jl::_cs_static_palindrome_subcycle_count, _palindrome_subcycle_count
CS multi-tracer fused kernels (X / Y / Z)	src/Operators/Advection/multitracer_kernels.jl
CS paired split seam exchange	src/Operators/Advection/CubedSphereSeams.jl
CS paired split seam adjoint	src/Adjoints/CubedSphereSeams.jl
CS panel-edge halo sync	src/Grids/PanelConnectivity.jl + cs_native_fluxes.jl::_propagate_cs_outflow_to_halo!
Lin-Rood cross-term + del-2 damping	src/Operators/Advection/linrood_horizontal.jl (driver), LinRood.jl (kernels, damping)

What's next

• Conservation budgets — the synthetic-fixture tests that verify the conservation contract bit-by-bit.

• Validation status — what we've actually validated end-to-end vs what's still on the to-do list.