Numerical Methods on Harry's Personal Site

Limited Variational Reconstruction: contacts, weak shocks, and the cost of a high-order target

Mon, 19 Oct 2026 14:20:00 +0800

The earlier LVR v2 post introduced a selective penalty that stabilizes high-order reconstruction near shocks. This follow-up brings together the subsequent research: a density-sensitive branch for steep contacts, an experimental weak-shock branch, completed two-dimensional comparisons, and a trial of high-order penalty targets. The main conclusion is a trade-off, not a single method that wins every case.

The synthesis here reflects results available through October 19, 2026. In the completed studies, the limited second-order (O2) target remains the reference choice. A newer high-order target improves several one-dimensional profiles but has not retained the O2-target method’s convergence advantage.

The common reconstruction idea

Let $c_i$ contain the nonconstant polynomial coefficients reconstructed in cell $i$, and let $c_i^{O2}$ be a limited O2 polynomial embedded in the same basis. LVR adds a cell-local penalty toward that target:

$$ \frac{\lambda_i}{2}(c_i-c_i^{O2})^T A_i(c_i-c_i^{O2}), \qquad \alpha_i=\frac{\lambda_i}{1+\lambda_i}\in[0,1]. $$

Here $A_i$ is the existing local variational-reconstruction block. For a fixed target and gate, its block update is a blend of the ordinary variational update and $c_i^{O2}$, with weights $1-\alpha_i$ and $\alpha_i$. The gate is built from the evolving cell means, but the reconstruction solve remains linear in the unknown high-order coefficients. At zero gate the original high-order reconstruction is recovered; at unit gate the local update returns the O2 target. This is a penalty inside reconstruction, not a limiter applied afterward.

The successive versions differ mainly in where they apply that blend:

Version	Added sensing mechanism	Main lesson
Initial LVR	Pressure jump and compression	Selective limiting can stabilize the shock while retaining high order elsewhere.
LVR v2	Supported pressure-extremum correction	Repairs a weakly gated shoulder beside a strong shock.
LVR v3, also called Average H2	Separate density-extremum correction	Handles steep advected contacts much better, with cap-dependent trade-offs.
Experimental weak G1	Pressure-gradient consistency and compression	Suppresses isolated weak-shock ringing in 1-D when held within each implicit stage.

The familiar v2 Lax result remains a useful point of departure: its wrong-direction pressure excursion fell from 0.11452 in the matched v1 run to 0.006859, while the density $L_1$ error changed only slightly (0.004208 to 0.004284). The correction addresses a local defect rather than uniformly lowering every error norm.

The v1/v2 Lax comparison shows why a shock sensor needs support beyond its strongest face: v2 suppresses the spurious pre-shock shoulder while preserving the main discontinuity. Curves are evolved cell averages, not reconstructed point values.

Steep contacts need a different signal

A pressure/compression gate can be quiet at a density contact. The v3 branch therefore measures the normalized density jump across faces and uses nearby cell-mean density extrema to support a narrow correction. Its shock and contact penalties are combined by

$$ \alpha_i=\max(\alpha_{s,i},\alpha_{c,i}), $$

where $s$ and $c$ denote the shock and density branches. Two neighbour-expansion passes act on accepted density-extremum seeds; the whole contact region is not expanded indiscriminately. The two caps quoted below are ordered as (shock, contact).

In the advected 10:1 contact, the mixed-cap v3 run $(0.5,1)$ reduces global density overshoot from about 0.733 for shock-only v2 to 0.03395. This does not make v3 universally preferable: the same mixed-cap series increases Toro 123 near-vacuum density $L_1$ error from 0.004974 to 0.006655. Stronger density treatment also has a width cost. A three-pass contact variant is more bounded in the contact tests but spreads the correction farther.

The top panels test whether the density correction disturbs Lax, Sod, or two-shock regions; the lower panels show advected 2:1, 10:1, and 100:1 contacts. Mean-supported v3 markedly reduces ringing at steep contacts. The WENO curves are complete-scheme context, not a reconstruction-only comparison, because their flux and time integrator differ.

The later cap study retained $(0.5,0.5)$ and $(1,1)$ as candidates. For the 10:1 contact, their overshoots are 0.06090 and 0.03395, respectively. On the fixed-budget Mach-20 cylinder, the stronger shock cap improves the final absolute density residual from $6.90\times10^{-5}$ to $8.38\times10^{-8}$, although its nearest-wall density moves slightly farther from the ideal stagnation-point value. Larger caps also increase some discontinuity-dominated errors, notably in Toro 123 and Shu–Osher. The historical mixed pair remains evidence, not the selected next setting.

The steady cylinder comparison isolates an important trade-off: the unit-cap v3 case drives the residual much lower within the same iteration budget, but the near-wall density profile is not uniformly improved. A low residual and a locally accurate wall value are different tests.

What survives in two dimensions?

Completed double-Mach-reflection and four-state Riemann runs show that v3 can stay selective while retaining resolved two-dimensional structures. On the diagonally symmetric 246U4S Riemann mesh, all six tested methods reached $t=0.8$ with finite positive density and pressure. V2 and v3 $(0.5,0.5)$ are almost indistinguishable in the final density view. V3 $(1,1)$ changes the roll-up pattern and reduces the measured density symmetry defect relative to v2, without flattening the flow to the smoother O2 result.

246U4S method	Final density min / max	Density symmetry $D_1$	Elapsed time
O2/WBAP	0.13365 / 1.77897	$2.48\times10^{-6}$	00:43:24
Full P3 3WBAP	0.12972 / 1.80020	$1.244\times10^{-3}$	03:04:05
Full P3 CWBAP	0.13019 / 1.81093	$7.36\times10^{-4}$	02:47:21
LVR v2 $(0.5,0)$	0.12714 / 1.78644	$1.056\times10^{-4}$	01:54:26
LVR v3 $(0.5,0.5)$	0.12714 / 1.78644	$1.064\times10^{-4}$	01:54:55
LVR v3 $(1,1)$	0.13550 / 1.77750	$4.91\times10^{-5}$	01:55:59

The symmetry measure compares reflected cells across $x=y$; it is not error against an exact solution. Likewise, density extrema and elapsed time do not by themselves establish an accuracy or hardware-speed ranking. The LVR rows took roughly 1 h 55 min versus 2 h 47 min–3 h 04 min for the fully limited P3 rows in these separate runs, partly because the latter needed more implicit iterations.

Cell-average density at $t=0.8$, shown over the same central $x,y$ window and color scale for all six methods. The top row contains O2, 3WBAP, and CWBAP; the bottom row contains v2 and the two symmetric-cap v3 settings. The v3 unit-cap result remains structured but differs visibly from the lower-cap roll-up. These visual differences are not an exact-solution ranking.

A weak-shock extension—and its temporal caveat

The ordinary v3 gates can remain nearly zero at isolated Mach-1.01–1.10 shocks, leaving density overshoot at roughly 8–15% of the exact shock jump. The experimental G1 branch asks whether the pressure change between cell means is consistent with unlimited O2 pressure-gradient extrapolations to their shared face. It also requires compression. The weak penalty is added by a maximum with the established shock and contact branches, so it does not replace them.

Updating this sensitive gate at every nonlinear inner iteration produced many iteration-cap hits. The tested policy evaluates it freshly at each ESDIRK stage, then holds only this branch fixed during that stage’s remaining inner evaluations. With that policy, overshoot at Mach 1.05 falls from 15.015% of the jump to 0.03925%, and the 428 cap hits of the every-inner-update run fall to zero. Across Mach 1.01–1.10, the frozen-gate overshoot stays below 0.04% of the jump in these tests. The stronger stabilization broadens the front; full 3WBAP still has slightly smaller density errors in all four isolated-shock rows.

The top row shows isolated Mach-1.02 and Mach-1.05 shock profiles; lower rows check Lax/Sod, steep contacts, and smooth entropy/acoustic waves. The nearly overlapping curves compare G1 refreshed every inner iteration with G1 frozen within each stage. Similar saved profiles hide a large difference in inner-iteration cap hits; the controls do not establish multidimensional behavior.

The sensor is a consistency test for a low-order pressure model, not a proof that a physical shock is present: smooth cubic pressure can produce a nonzero mismatch. Stage freezing also changes the nonlinear temporal coupling. The weak branch remains optional and off by default; its multidimensional behavior and time-step-refined accuracy are open questions.

Testing a high-order target

The next experiment replaces the O2 penalty target with a limited full-degree predictor. For each flow right-hand-side evaluation, an ordinary unpenalized reconstruction sweep produces a predictor $q$; selected cells receive a 3WBAP- or CWBAP-limited target $t$, while the LVR gate still comes from the WBAP-limited O2 sensor. The penalized update becomes

$$ r_i^{m+1}=(1-\omega_i)r_i^m+\omega_i\big[(1-\alpha_i)G_i(r^{\mathrm{used}})+\alpha_i t_i\big]. $$

The target is frozen for that reconstruction solve but rebuilt at the next flow evaluation. This keeps the local algebra linear for a fixed target while feeding the incoming high-order coefficients back into the outer process. It is therefore a different, more tightly coupled experiment—not a replacement that can inherit the O2-target results.

All 49 planned one-dimensional/steady rows reached their declared horizon or fixed budget. Some high-order targets improve Lax and Shu–Osher density norms and preserve more of the latter’s wave train: its global density $L_1$ changes from 0.033835 for the O2-target v3 control to 0.022324 with a 3WBAP target and 0.018117 with a CWBAP target. But local behavior differs: the ordinary 3WBAP target increases advected 10:1 contact overshoot from 0.033945 to 0.10802, whereas a wider CWBAP target reduces it to 0.0010012. These are case-specific outcomes, not a monotonic trend with target order.

The selected Shu–Osher wave window illustrates why a high-order target is attractive: several variants retain more post-shock wave structure than the O2-target LVR control. Their global norm alone cannot determine whether local shock and contact excursions are acceptable.

The decisive counterexample is the fixed-budget cylinder. Its O2-target LVR run ends at an absolute density residual of $8.3840\times10^{-8}$. The 3WBAP- and CWBAP-target variants end at 0.079925 and 0.065909; widening the target bands only improves those to 0.035074 and 0.029043. Some wall values look closer to the theoretical stagnation point, but the steady fields have not converged comparably.

At the same 5000-iteration budget, the high-order-target profiles may look locally appealing while their residual histories remain far above the O2-target run. The plot is a convergence warning, not a cost-to-convergence comparison.

The available two-dimensional extension strengthens that warning but is incomplete as of this synthesis. Both high-order-target DMR runs reached a 24-hour timeout before $t=0.25$; the wider-band run completed 283/625 steps and hit its inner cap in 99.51% of logged stages. The normal-band run completed 520/625 steps with 29.04% capped stages. The 246U4S normal-band case was still running at 896/3200 steps, with 99.82% of logged stages capped; three other rows were pending. Thus there is no accepted final-time two-dimensional result for this target experiment.

These histories show numerical work, not physical accuracy: the wider DMR target advances fewer steps despite the same wall-time limit because nearly every implicit stage exhausts its inner-iteration budget. Partial contours from these runs cannot be treated as converged final solutions.

Current reading of the evidence

The O2-target LVR family offers useful selective stabilization: v2 repairs the strong-shock shoulder, v3 gives a much better steep-contact option, and frozen G1 closes an isolated one-dimensional weak-shock gap. None is universally superior. Toro near-vacuum behavior, cap-dependent two-dimensional structures, and the weak sensor’s smooth-field response remain real limits. The experimental high-order target produces interesting waves and some lower one-dimensional norms, but its steady and implicit-transient convergence losses are too large to recommend it as the new default. A future target design must preserve those convergence properties, not just improve selected pictures.

Reaction-region indicators revisited: the cross-case tradeoff

Mon, 19 Oct 2026 13:30:43 +0800

The first RRI progress post introduced the selector, and the September comparison set out broadA, BA-CE6, and Balance. This installment follows the research through 19 October 2026: a consolidated flame matrix, the local-rate-ratio candidates, a refined two-scale comparison, and a completed Kinetic-response campaign. It draws on existing runs; preparing the synthesis did not launch new CFD.

The result is a useful mixed-method demonstration, not a universal new default. localR10 gives the clearest interface/ignition compromise at coarse steps, but loses broadA’s H2/O2 flame robustness. Kinetic response largely recovers flame behavior, yet does not reproduce the desired two-scale compromise.

What is being selected?

The cellwise switch $\chi$ partitions chemistry between an implicit flow step and two local-chemistry half-steps. $\chi=0$ is fully coupled; $\chi=1$ is full Strang splitting. Intermediate values split only part of the chemical source. The selector is evaluated at the start of each physical step and held through that step, then recomputed on the evolved state. A selector image beside a terminal profile usually shows the last held field, not a new evaluation at the endpoint.

The comparisons below use a-posteriori trajectories when discussing speed or temperature error. A-priori maps on saved states show what a recipe would select, not what its evolved solution would do. Actual physical steps, inner-iteration caps, and the chosen numerical reference remain separate qualifications.

The expanded flame matrix

The corrected, nitrogen-diluted H2/O2 flame has a Cantera burning-speed reference of 2.2539659772 m/s. The new curation puts eight available treatments on a common three-ODE, three-step view: 72 comparison cells, 57 completed and 15 failed. Every completed recent-campaign trajectory reached its 100-iteration inner cap at every physical step. These are fixed-work fitted front speeds, not fully converged flame solutions; a dagger below marks $R^2\lt0.9$.

ODE / code step	Coupled	Strang	broadA	BA-CE6	localR10	Kinetic
ESDIRK2 / 0.002	−1.001%	+13.931%	−1.112%†	−1.112%†	+4.808%	−1.014%
ESDIRK2 / 0.004	−1.235%	failed	−1.119%	−1.119%	failed	+0.262%
U2R1 / 0.004	+38.652%†	failed	+16.707%†	+7.681%†	failed	failed
U2R2 / 0.004	−1.911%	failed	−1.831%†	−1.885%†	failed	−1.190%

These selected rows explain the cross-case tension. broadA and BA-CE6 retain the roughly −1% ESDIRK2 flame-speed behavior when Strang’s fitted error grows or its run fails. localR10 cannot keep that protection. Kinetic is promising on ESDIRK2 at the largest step and completes U2R2, but its largest-step U2R1 run fails. The coarse U2R1 fitted values in the table all have poor linearity; a smaller percentage there does not establish a steady accurate flame. Balance without spatial expansion has no flame posterior matrix and should not be inferred from B-E2.

ESDIRK2 flame comparison. Each point is a signed fitted-speed error against the Cantera reference at one code step. The broadA and BA-CE6 curves overlap. Crossed markers identify reported poor fits; missing Strang and localR points at the largest step are failed runs, not zero errors. Lines join measured points, not a convergence law.

U2R1 flame comparison. The plot retains the large coarse-step errors and failures on its full vertical scale. Several fitted front speeds are nonstationary, so the apparent ordering of their percentages is weaker evidence than completion and fit quality. Kinetic has no largest-step point because that run failed.

U2R2 flame comparison. Strang and the localR variants fail across the tested steps; Kinetic returns coupled-like fitted errors. broadA and BA-CE6 complete, but their medium- and large-step fits have low linearity. B-E2’s completion does not imply improved speed error.

Detonation gives a complementary check rather than a single winner. Against the same-grid time-extrapolated speed of 2798.225 m/s, the corrected fine-grid matrix at code step $8\times10^{-7}$ gives:

ODE	Coupled	Strang	broadA	localR10
ESDIRK2	+1.295%	+0.311%	+0.589%	+0.311%
U2R2	+0.360%	+0.943%	+1.017%	+0.855%

The preferred endpoint reverses with the ODE method. Spatial convergence of that reference is still deferred. BA-CE6, Balance, B-E2, and Kinetic have no detonation a-posteriori matrix; a selector contour cannot supply those missing outcomes.

The two-scale result: localR10 helps, but at particular steps

The n-heptane case separates a narrow reacting interface near 2 mm from a sensitive ignition region farther downstream. All compared short-window trajectories restart from the same state and end at 32 microseconds. Temperature errors are measured without shifting either front: interface means/maxima use $1\le x\lt3$ mm, and ignition uses $x\ge3$ mm.

The refined comparison uses a Strang reference at code step $2.5\times10^{-5}$. Refining the earlier reference by a factor of two changed interface mean/max errors by 3.40/78.31 K and ignition mean/max errors by 1.34/60.97 K. Those differences qualify small gains; they are not rigorous error bounds or proof that the Strang reference is exact.

Code step	Treatment	Interface mean / max [K]	Ignition mean / max [K]	Implicit stages converged / capped
0.0004	Coupled	21.11 / 370.87	44.71 / 686.62	16 / 0
0.0004	Strang	22.59 / 651.14	12.24 / 361.96	16 / 0
0.0004	broadA	21.53 / 443.67	14.74 / 317.39	14 / 2
0.0004	localR10	20.81 / 412.76	10.56 / 242.77	16 / 0
0.0008	Coupled	failed	failed	1 / 3
0.0008	Strang	12.28 / 538.32	30.50 / 574.34	7 / 1
0.0008	broadA	15.36 / 238.02	77.29 / 1049.96	7 / 1
0.0008	localR10	8.75 / 149.31	30.39 / 571.17	7 / 1

At $4\times10^{-4}$, localR10 has roughly coupled-like interface error and Strang-like ignition error, with all stages converged. Its 0.30 K interface-mean advantage over coupled is smaller than the reference-refinement sensitivity; its ignition-mean improvement over coupled is much larger. At $8\times10^{-4}$, it substantially reduces Strang’s worst interface error while keeping similar ignition error, but all completed methods have a capped stage and coupled has no endpoint. At the smaller $10^{-4}$ step, Strang has the best interface mean among these recipes. A region does not have one permanent preferred method.

Two-scale matrix against the refined reference. The upper panels show interface mean and maximum absolute temperature error; the lower panels show ignition mean and maximum. The horizontal axis is the code step in units of $10^{-4}$. Kinetic appears only at its two matched smaller steps. Missing points, including failed coupled at the largest step, are not zeros; connecting lines do not establish an order of convergence.

The localR10 choice uses chemical and estimated molecular-transport rates in shared coordinates $q=(Y_1,\ldots,Y_{N_s},\ln(T/T_*))$. Write their rate vectors as $s$ and $d$, with $\mathcal A=\|s\|_2$ and $\mathcal D=\|d\|_2$. Its coupled share is

$$ f_{R10}=g_h H(\Delta t\mathcal A;0.001,1) H(\mathcal D/\mathcal A;0.10,4), \qquad \chi_{R10}=\mathcal P_{0.10,0.05}(1-f_{R10}). $$

Here $H(z;z_0,p)=(z/z_0)^p/[1+(z/z_0)^p]$, and $g_h$ is the existing pressure-jump gate. The endpoint map snaps $\chi\le0.10$ to coupled and $\chi\ge0.95$ to Strang. Zero chemical or diffusive rate selects Strang. Unlike broadA’s gradient-timescale proxy, this ratio compares estimated rates; it still omits parts of the flow residual and is not an error estimator.

Tuning is not transferable by inspection. These interface-mean errors use the earlier Strang reference at $5\times10^{-5}$, not the refined reference used in the main table. localR10P2 is attractive at code step 0.0004 but loses that advantage at 0.0008; B-E2’s wider selected interface does not improve both interface metrics. The two plotted step sizes do not define a convergence slope.

Same-start trajectories at code step 0.0004. The top panels enlarge interface and ignition temperatures at the common endpoint; the lower panels show signed differences from the earlier $5\times10^{-5}$ Strang reference. This is where localR10P2 looks especially favorable. The reference’s zero-error trace is self-comparison, not exact physical truth, and these plotted errors must not be subtracted from the refined-reference table.

Same-start trajectories at code step 0.0008. broadA limits one interface peak but permits larger ignition excursions. localR10 preserves Strang-like ignition behavior while improving the interface; the P2 variant loses its smaller-step advantage. Coupled failed and therefore has no endpoint curve. Several completed rows have a capped stage.

Selectors on their own evolved states. The figure pairs final temperature with each recipe’s last held selector at code step 0.0008. It shows nontrivial local mixing without spatial expansion, but it is not a same-state causal comparison: each recipe also changed the state from which its next selector was computed.

Kinetic response: a useful diagnostic, not the missing default

The Kinetic attempt asks how strongly estimated transport changes the chemical rate, relative to chemistry changing its own rate. With the same rate vectors $s,d$ and a constant-density chemical Jacobian $J_s$, it uses

$$ R_K=\frac{\|J_s d\|_2}{\|J_s s\|_2}, \qquad \chi_K=1-H(R_K;0.1,2). $$

This is a local directional-response ratio, not the full spatial splitting error. A zero numerator, including the both-zero case, selects Strang; a positive numerator with zero denominator selects coupled. The implemented campaign uses constant-volume chemistry, no activity or shock gate, and no nonzero snapping tolerance. Its ratio has no explicit timestep dependence on a fixed state, although the evolving states still depend on the step. On the corrected flame, Kinetic largely restores coupled-like fitted speeds, including a +0.262% ESDIRK2 error at the largest step. The U2R1 run at that step still fails.

At the matched two-scale code step 0.0002, Kinetic’s ignition mean/max errors are 8.35/182.83 K. Strang gives 3.82/114.23 K, and broadA 3.96/62.29 K. Kinetic’s interface mean is 27.76 K, above coupled’s 21.48 K. It does not reproduce localR10’s later coarse-step compromise. Its requested 0.0004 and 0.0008 ceilings were both clipped by CFL to the same 14-step history, with a median actual step of about 0.0002305. Those results are not equal-large-step comparisons with the eight- and four-step controls.

Matched two-scale selection at code step 0.0002. Temperature appears above the last held broadA and Kinetic selectors. Zero is coupled and one is Strang. Kinetic retains more coupled chemistry through the ignition region; that accompanies its larger ignition error, but the plot alone cannot establish causality.

Native solver fields, not a rounded-state reconstruction. The H2 flame at code step 0.002 and two-scale n-heptane at 0.0002 pair endpoint temperatures with last-step selectors. Kinetic activates irregularly even in nearly isothermal regions. A satisfactory flame-speed fit does not establish that this cellwise field is smooth or well conditioned.

The fixed-state mapping screen explains part of the difficulty. With $H(R_K;0.1,2)$, the worst reference-interface coupled coverage is 0.674 while worst reference-ignition coupling is 0.421. Moving the midpoint to 0.03 raises interface coverage to 0.865, but also raises ignition coupling to 0.601. For the report’s illustrative 0.8-interface/0.1-ignition targets, even an optimized monotone remapping cannot satisfy every reference state. Those targets diagnose the design tradeoff; they are not physical truth labels.

The remapping limit is visible before a posterior run. This fixed-state screen compares interface preservation with ignition release as the Kinetic ratio is remapped. Overlap between states forces a tradeoff for the stated diagnostic targets. The plot is not a graph of CFD temperature error and cannot replace the matched posterior result.

What remains open

broadA remains the strongest practical cross-case mixed baseline in these tests; localR10 is the strongest demonstration of selective interface/ignition behavior at the coarse two-scale steps. Neither combines both advantages across cases. BA-CE6 preserves the tested flame but selects Strang throughout the sampled two-scale states, with no separate two-scale posterior row. B-E2, stronger snapping, and the Kinetic response each improve some measurements while losing others. No new 2-D propagation or GRI30 mixed-method validation resolves the transfer question.

The next proposed diagnostic is a finite-time, transport-forced local reactor probe: compare a splitting-defect proxy with the coarse coupled-chemistry integration error over the actual proposed step. This is a research proposal, not an implemented or validated selector. A decisive later test would still need matched actual steps, separate accounting of capped nonlinear stages, and reference checks across flame, detonation, interface, and ignition. The goal is not a prettier coupling contour; it is one interpretable rule that retains broadA’s flame protection and localR10’s two-scale compromise.

Reactive-region selection: broadA, BA-CE6 and Balance

Tue, 29 Sep 2026 12:42:02 +0800

Evidence curated 29 September 2026. This post brings together the corrected H2/O2 benchmarks and the n-heptane ignition-plus-front experiment. The numerical cutoff includes the completed Balance test at code step $4\times10^{-4}$. The $8\times10^{-4}$ comparison has reserved slots below; its numerical results are not included in this edition. No new CFD was launched during curation.

Conclusions and evidence coverage

broadA remains the established flame-preserving recipe. BA-CE6 reproduces its ESDIRK2 flame-speed behavior while allowing chemically difficult, weakly diffusive ignition states to select Strang. However, the archived flame matrices are fixed-inner-work calculations, not converged implicit solutions.
Detonation does not have a universal preferred endpoint. On the same fine grid, ESDIRK2 at $8\times10^{-7}$ favors Strang, while U2R2 at that step favors coupled. No BA-CE6 detonation a-posteriori matrix exists in the archive; similarity of its frozen selector to broadA is not a substitute for one.
The two-scale case exposes different regional errors. At matched $4\times10^{-4}$, coupled better preserves the interface’s worst temperature error than Strang, but has much larger ignition-kernel error. Balance retains localized interface mixing and improves kernel errors substantially relative to coupled. It does not win every metric, and its small mean-error advantage over Strang is below the available reference-refinement sensitivity.
BA-CE6 is too eager to escape in this particular prepared case. It selects Strang everywhere in all 55 evaluated snapshot/timestep combinations. Balance is a rate-balance alternative, not a validated universal replacement.

Evidence	broadA	BA-CE6	Balance
Corrected H2/O2 flame, 3 ODE × 3 steps, with endpoint controls	Complete terminal accounting	Same campaign: 31 completed, 5 failed across all 36 rows	Unrun
Fine H2/O2 detonation, 3 ODE × 4 steps, plus coarse ESDIRK2	Complete: 52 rows including endpoints/default	Unrun; a-priori only	Unrun
Earlier hot n-heptane, ESDIRK2 × 5 steps × 4 treatments	Complete	Complete	Unrun
Prepared ignition + front, same-checkpoint short window	Completed at $2\times10^{-4}$ and $4\times10^{-4}$	A-priori only; not a separate CFD row	Completed at $4\times10^{-4}$
Prepared ignition + front, $8\times10^{-4}$	Pending integration	Not scheduled	Pending integration

“Complete accounting” includes failures; it does not mean every row yields a valid accuracy result. The detailed two-scale report contains all comparison plots, tables, startup records and frozen-state figures. The linked study records require access to the research repository; the figures needed to read this post are included on this page.

Common mixed-integration contract

Let $U$ be the cell-mean conservative state, $F(U)$ the spatial flow residual (convection and diffusion, with nonchemical sources if present), and $S(U)$ the local chemical source. For a field $\chi_i\in[0,1]$ frozen at the beginning of each physical step, partition the evolution as

$$ \frac{\mathrm dU}{\mathrm dt}=F(U)+S(U),\qquad G_\chi(U)=F(U)+(1-\chi)S(U),\qquad Q_\chi(U)=\chi S(U). $$

Advance $Q_\chi$ for half a step, $G_\chi$ for a full step, and $Q_\chi$ for half a step. The cell-local chemistry advance is constant-volume; the flow advance uses the selected ODE method. Thus $\chi=0$ is fully coupled and $\chi=1$ is full Strang. Values strictly between them split only part of the chemistry. This is not a convex combination of two completed solutions. The separate debugging source multiplier multiplies chemistry independently; it is unity in these studies. Nonchemical sources are not multiplied by $\chi$.

All three retained recipes use the same snapping operator and zero spatial passes:

$$ \mathcal P(z)= \begin{cases} 0,&z\le0.01,\\ z,&0.01\lt z\lt0.95,\\ 1,&z\ge0.95. \end{cases} $$

Exact zero skips that cell’s split chemistry advance; exact one removes its chemistry work from the implicit flow RHS. The upper threshold here is the retained recipe’s 0.05 distance from one, not the general solver default. Post-step diagnostic output contains the last pre-step frozen selector, not necessarily a selector reevaluated on the plotted final state.

The common threshold-and-power function is

$$ H(z;z_0,p)=\frac{(z/z_0)^p}{1+(z/z_0)^p},\qquad z\ge0,\quad z_0>0,\quad p>0. $$

$z_0$ is the half-activation value and $p$ controls transition sharpness. All arguments and thresholds below are dimensionless. The physical step is $\Delta t=\Delta t_{\rm code}L_0/U_0$; for the two-scale case, $L_0=1\,\mathrm m$, $U_0=379\,\mathrm{m\,s^{-1}}$.

Constituent fields used by broadA and BA-CE6

For species $k$, let $Y_k$ be mass fraction, $W_k$ molecular weight, $\dot\omega_k$ net molar production per volume per time, and $\dot m_k=W_k\dot\omega_k$ net mass production per volume per time. $\rho$ is mass density; $T$ is absolute temperature; $c_v$ is mixture mass-specific constant-volume heat capacity; $h_k$ is species mass-specific enthalpy including formation enthalpy. Define

$$ \dot q_h=-\sum_k h_k\dot m_k,\qquad T_{\rm scale}=\max(T,T_{\rm floor}),\qquad A=\left[\sum_k\left(\frac{\dot m_k}{\rho}\right)^2+ \left(\frac{\dot q_h}{\rho c_vT_{\rm scale}}\right)^2\right]^{1/2}. $$

$\dot q_h$ is an enthalpy-based volumetric heat-release proxy in $\mathrm{W\,m^{-3}}$, not an exact constant-volume temperature derivative. $T_{\rm scale}$ is a temperature in kelvin; the ratio in $A$ has units $\mathrm{s^{-1}}$. $T_{\rm floor}$ is the mechanism’s supported lower temperature bound. There is no logarithm of a dimensional temperature.

For face-neighbor cells $i,j$, let $\ell_{ij}$ be physical center distance, $\ell_i$ the physical maximum cell-length scale, and $Y_*=10^{-3}$. The solver-form gradient estimate is

$$ G_i=\max_{j\in\mathcal N(i)}\left[ \left(\frac{T_j-T_i}{\ell_{ij}\max(T_i,T_j,T_{\rm floor})}\right)^2+ \sum_{k:\,\max(Y_{k,i},Y_{k,j})\ge Y_*} \left(\frac{Y_{k,j}-Y_{k,i}}{\ell_{ij}\max(Y_{k,i},Y_{k,j},Y_*)}\right)^2 \right]^{1/2}. $$$$ L_{{\rm grad},i}=\max(G_i^{-1},\ell_i),\qquad B_i=\frac{D_{\max,i}}{L_{{\rm grad},i}^{2}},\qquad a_i=\Delta t A_i,\quad b_i=\Delta t B_i. $$

Use $B_i=0$ when $G_i=0$. $D_{\max}$ is the largest mixture-averaged species diffusivity in $\mathrm{m^2\,s^{-1}}$. $B$ is an inverse-time diffusion-scale estimate, not an evaluated diffusive RHS. Physical-boundary faces without a neighbor do not add a jump. The temperature term approximates the gradient of $\ln(T/T_{\rm ref})$ for any fixed positive reference temperature; the finite difference is not exactly a logarithmic difference. The cell-length bound limits activation by gradients below the represented spatial scale.

The pressure sensor and shock suppression are

$$ h_i=\max_{j\in\mathcal N(i)} \frac{|p_j-p_i|}{\max(|p_i|,|p_j|,p_\epsilon)},\qquad g_{h,i}=\frac{1}{1+(h_i/0.08)^4}. $$

Here $p$ is pressure and $p_\epsilon$ is only a positive numerical safeguard, negligible in these positive-pressure cases. The symbol $h_i$ denotes a pressure jump, distinct from species enthalpy $h_k$. Strong jumps reduce coupling; this is an empirical shock preference, not a theorem that Strang is always more accurate in detonations.

broadA: broad activity activation

$$ C_A=H(a;10,1/2)H(b;2,1/2)g_h,\qquad f_A=H(C_A;0.005,3),\qquad \chi_A=\mathcal P(1-f_A). $$

$f_A$ is the raw coupled fraction. The square-root Hill factors activate gradually, and the low outer midpoint broadens the coupled band relative to the earlier default. No expansion/filter passes are used. $A$ and $B$ must both be nonzero. At fixed positive rates, $\Delta t\to0$ gives Strang; $\Delta t\to\infty$ gives raw coupled fraction $H(g_h;0.005,3)$. Hence a smooth, weakly diffusive but very stiff ignition region can eventually become strongly coupled even when that is not advantageous.

BA-CE6: broadA with chemistry escape

Define the concentration-Jacobian proxy

$$ \Lambda=\max_k\left|\frac{\partial\dot\omega_k}{\partial C_k}\right|, \qquad s=\Delta t\Lambda,\qquad r=\frac{B}{\Lambda}, $$

where $C_k$ is species molar concentration. The diagonal derivatives use Cantera’s net_production_rates_ddCi: temperature, pressure and all other species concentrations are held fixed under that API’s derivative convention. Cantera kinetics reference. This is not the spectral radius of the full constant-energy reactor Jacobian, nor a measured nonlinear convergence rate. $\Lambda$ has units $\mathrm{s^{-1}}$. For $\Lambda>0$, set

$$ E=H(s;100,2)\,[1-H(r;10^{-6},6)],\qquad \chi_{\rm CE6}=\mathcal P\bigl(1-f_A(1-E)\bigr). $$

Set $E=0$ when $\Lambda=0$. The gate releases chemistry toward Strang when the timestep is long relative to the fast chemical scale and diffusion is weak relative to that scale. It can only remove broadA coupling. “6” is the power on the $B/\Lambda$ gate, not an ODE order. At positive fixed rates,

$$ f_{{\rm CE6},\infty}=H(g_h;0.005,3)H(B/\Lambda;10^{-6},6). $$

This preserves the tested H2/O2 flame band but can erase a real interface if a very fast chemical mode drives $B/\Lambda$ below the threshold. The numerical value $10^{-6}$ is calibrated, not a universal physical boundary. This recipe is the refined mode-2 BA-CE6, not the earlier unsuccessful ratio-only v2.

Balance: compare actual chemical and diffusive rate coordinates

Use all species and temperature in the same coordinates:

$$ \mathcal A=\left[\sum_k\dot Y_{S,k}^{2}+(\dot T_S/T)^2\right]^{1/2}, \qquad \mathcal D=\left[\sum_k\dot Y_{D,k}^{2}+(\dot T_D/T)^2\right]^{1/2}. $$

Both norms have units $\mathrm{s^{-1}}$. They contain no division by trace species fractions. With species mass-specific internal energy $e_k(T)$, including formation energy,

$$ \dot Y_{S,k}=\dot m_k/\rho,\qquad \rho c_v\dot T_S=-\sum_k e_k\dot m_k. $$

Unlike broadA’s $\dot q_h$ proxy, this chemical temperature rate is consistent with fixed volume and total energy. For molecular diffusion, form

$$ \mathbf j_k^{0}=-\rho D_k\nabla Y_k,\qquad \mathbf j_k=\mathbf j_k^{0}-Y_k\sum_l\mathbf j_l^{0},\qquad \mathbf q=-\kappa\nabla T+\sum_k h_k\mathbf j_k, $$$$ S_{D,k}=-\nabla\cdot\mathbf j_k,\qquad \dot Y_{D,k}=S_{D,k}/\rho,\qquad \rho c_v\dot T_D=-\nabla\cdot\mathbf q-\sum_k e_kS_{D,k}. $$

$\kappa$ is thermal conductivity; $\mathbf j_k$ is a mass flux with zero summed species flux; $\mathbf q$ includes conduction and diffusing species enthalpy. The last subtraction removes composition-induced internal-energy change before converting the energy rate to a temperature rate. In the diagnostic, centered two-point cell-mean gradients and averaged face coefficients estimate these divergences. Physical-boundary diffusive flux is zero. This estimator does not replace the solver’s reconstructed spatial residual and does not include convection, compression or viscous heating.

$$ \alpha=\Delta t\mathcal A,\qquad d=\Delta t\mathcal D,\qquad Z=\frac{\alpha d}{(1+\alpha)^2},\qquad \chi_{\rm Balance}=\mathcal P\left(\frac{1}{1+(g_hZ/\theta)^p}\right), \qquad \theta=0.001,\quad p=2. $$

$\alpha$ is deliberately distinguished from broadA’s $a$: its temperature component uses internal energy rather than enthalpy. The midpoint and power were selected from an a-priori screen before this candidate’s CFD result: $\theta\in\{0.0005,0.001,0.002,0.005,0.01,0.02\}$, $p\in\{2,3\}$. There is no Jacobian, chemistry-escape ratio or additional activity gate. The shared shock gate and endpoint thresholds remain parameters.

For fixed nonzero $\mathcal A$,

$$ Z\sim\Delta t^2\mathcal A\mathcal D\quad(\Delta t\to0),\qquad Z\longrightarrow\mathcal D/\mathcal A\quad(\Delta t\to\infty). $$

Zero chemistry or diffusion selects Strang. Large-step coupling therefore requires an appreciable current diffusive rate relative to chemical rate, not merely a large gradient multiplied by an arbitrarily large timestep. Net-rate cancellation, stiff near-equilibrium chemistry, omitted convection, mesh dependence and physical boundary modeling remain blind spots. The implementation has only been validated on this orthogonal, closed strip.

H2/O2 flame: complete latest four-treatment matrix

This is the BA-CE6 campaign, with its own rerun broadA and endpoint controls, rather than selectively combining rows from different campaigns. It uses the corrected multispecies temperature gradient, a Cantera stationary initial profile, fresh BCIn and burned BCFar, $25\,\mu\mathrm m$ spacing, P1 FV, and final code time 0.5. The reference is $S_{L,\rm ref}=2.2539659772\,\mathrm{m\,s^{-1}}$. Burning velocity is recovered from fresh-gas velocity minus fitted laboratory-frame front speed:

$$ S_L=u_{\rm fresh}-\frac{\mathrm dx_f}{\mathrm dt},\qquad \varepsilon_S=100\left(\frac{S_L}{S_{L,\rm ref}}-1\right)\%. $$

Fits use code times 0.2–0.5. ODE codes are 204 (ESDIRK2), 412 (U2R1) and 411 (U2R2). All 36 requested rows have terminal accounting: 31 completed and five Strang failures. All 11,212 physical steps of the completed rows hit the 100-iteration cap; 15,255 observed implicit-stage groups are capped. These are fixed-work observations. An asterisk below marks $R^2<0.9$, where the fitted speed is only an interval average, not a reliable steady speed.

ODE	Code dt	Coupled	Strang	broadA	BA-CE6
ESDIRK2	0.00075	-1.223%	+1.259%	-1.186%	-1.186%
ESDIRK2	0.002	-1.001%	+13.931%	-1.112%*	-1.112%*
ESDIRK2	0.004	-1.235%	failed	-1.119%	-1.119%
U2R1	0.00075	-1.050%	-0.979%	-1.218%	-1.218%
U2R1	0.002	+2.680%*	+3.466%	+2.107%	+2.107%
U2R1	0.004	+38.652%*	failed	+16.707%*	+7.681%*
U2R2	0.00075	-1.025%	failed	-1.075%	-1.075%
U2R2	0.002	-1.268%	failed	-1.391%*	-1.572%*
U2R2	0.004	-1.911%	failed	-1.831%*	-1.885%*

ESDIRK2 flame errors. broadA and BA-CE6 retain roughly −1.1% fitted error; Strang reaches +13.9% at the middle step and fails at the largest. All completed curves are inner-capped, including those close to the reference speed.

U2R1 flame errors. The coarse-step BA-CE6 interval fit is closer to the reference than broadA or coupled, but all those coarse trajectories have low $R^2$; the smaller apparent percentage is not proof of a steady accurate flame.

U2R2 flame errors. Adaptive rows finish where Strang fails. Small signed errors in the medium/coarse adaptive rows coexist with low front-fit linearity.

Do not interchange the older broadA campaign

The earlier full matrix compared coupled/Strang/default/broadA. Its broadA coarse U2R1 error is +3.717%, whereas the later BA-CE6 campaign’s rerun broadA gives +16.707%. Their $R^2$ values are only 0.175 and 0.230. The later BA-CE6 value is +7.681% with $R^2=0.248$. All are fixed-work, nonstationary fits. This report uses the later four-treatment campaign consistently and does not replace its broadA control with the more attractive older number. The difference has not been isolated into a unique cause.

H2/O2 detonation: fine-grid posterior matrix

The laboratory-frame fitted ZND setup uses 5000 cells over 12.5 mm, $\Delta x=2.5\,\mu\mathrm m$, initial shock at 2.5 mm and left BCFar matching the downstream state. The fitted induction length is about $50.04\,\mu\mathrm m$ (about 20 cells). The same-grid reference is the observed-order time-limit extrapolation of the coupled ESDIRK4 study,

$$ D_0=2798.224963262598\,\mathrm{m\,s^{-1}},\qquad \varepsilon_D=100\left(\frac{D}{D_0}-1\right)\%. $$

This is not the equilibrium Cantera/SDToolbox CJ value $2836.381883789159\,\mathrm{m\,s^{-1}}$ and is not a spatially converged continuum reference. Spatial convergence remains deferred. All 52 archived rows have fitted speeds; front fits use $4.0\le x_f\le8.25$ mm and have $R^2\ge0.999978$. The default column is the earlier default-Hill control, not BA-CE6 or Balance. Every BA-CE6 entry is explicitly unrun.

ODE	Code dt	Coupled	Strang	default	broadA	BA-CE6
ESDIRK2	1e-07	+0.030%	+0.027%	+0.027%	+0.027%	unrun
ESDIRK2	2e-07	+0.135%	+0.073%	+0.073%	+0.076%	unrun
ESDIRK2	4e-07	+0.456%	+0.242%	+0.242%	+0.244%	unrun
ESDIRK2	8e-07	+1.295%	+0.311%	+0.420%	+0.589%	unrun
ESDIRK2	1.6e-06	+16.627%	+19.380%	+19.915%	+20.039%	unrun
U2R1	1e-07	+0.031%	+0.031%	+0.031%	+0.031%	unrun
U2R1	2e-07	+0.134%	+0.125%	+0.125%	+0.118%	unrun
U2R1	4e-07	+0.319%	+0.313%	+0.309%	+0.296%	unrun
U2R1	8e-07	+0.745%	+0.693%	+0.712%	+0.735%	unrun
U2R2	1e-07	-0.376%	-0.371%	-0.371%	-0.371%	unrun
U2R2	2e-07	-0.183%	-0.313%	-0.313%	-0.316%	unrun
U2R2	4e-07	-0.049%	-0.092%	-0.092%	-0.104%	unrun
U2R2	8e-07	+0.360%	+0.943%	+1.026%	+1.017%	unrun

ESDIRK2 through $8\times10^{-7}$. This zoom resolves the favorable Strang and broadA errors at useful timesteps without compressing them against the coarse-step breakdown. The complete larger-step view follows.

ESDIRK2 detonation errors. At $8\times10^{-7}$, Strang and broadA improve on coupled. The $1.6\times10^{-6}$ extension makes every treatment poor and dominates the vertical scale; use the table to resolve the smaller-step errors.

U2R1 detonation errors. Differences among treatments are much smaller than the timestep trend, so no large mixed-method accuracy advantage follows.

U2R2 detonation errors. At $8\times10^{-7}$ the endpoint ordering reverses: coupled has +0.360% error versus +0.943% for Strang and +1.017% for broadA. A shock preference alone cannot guarantee the better temporal method.

The available BA-CE6 detonation evidence is static. At the three annotated prospective steps, its selector overlaps broadA on this saved profile. This does not establish BA-CE6 speed, nonlinear convergence or runtime. Also note the convention: near-Strang means $\chi$ near one, not zero.

Why n-heptane and what NTC means

The mechanism has the Seiser/Pitsch/Curran reduced LLNL lineage. LLNL’s short mechanism descends from its older version-38 detailed mechanism, with 159 species and 770 reversible reactions; it is not the latest detailed LLNL mechanism. LLNL mechanism description. The actual DNDSR YAML used here contains 160 species and 1540 directed reactions, as counted in the saved Cantera manifest. It has the existing low-temperature thermodynamic extension, with constant endpoint heat capacities below the original interval, to support the solver’s 1 K bound. This numerical extension is not validation of reaction kinetics at cryogenic temperatures. All reductions use the exact species order; the absorbed last species in this case is c5h9o1-4, not nitrogen.

Chemicals represented

n-Heptane is the straight-chain saturated hydrocarbon $\mathrm{CH_3(CH_2)_5CH_3}$, with molecular formula $\mathrm{C_7H_{16}}$; its mechanism identifier is nc7h16. In this case it is gaseous fuel, not a liquid spray. The fresh oxidizer is oxygen diluted by nitrogen. The initial molar ratio is n-heptane : oxygen : nitrogen = 1 : 11 : 41.36, corresponding to the stoichiometric mixture. The overall complete-oxidation balance, not an elementary reaction in the kinetic mechanism, is

$$ \mathrm{C_7H_{16}}+11\,\mathrm{O_2} \longrightarrow 7\,\mathrm{CO_2}+8\,\mathrm{H_2O}. $$

The model does not jump directly from fuel to final products: it evolves small radicals, fragments and oxygenated intermediates through many reactions. The following groups explain the fields plotted in this study and the low-temperature pathways relevant to ignition:

Group	Representative exact identifiers	Meaning in this study
Fuel and oxidizer	`nc7h16`, `o2`	Gaseous n-heptane and molecular oxygen.
Bath gas	`n2`	Nitrogen diluent; contributes to mixture thermodynamics/transport and collision-partner effects, but is the only nitrogen-bearing species.
Major products and smaller molecules	`co2`, `h2o`, `co`, `h2`, `ch4`	Carbon dioxide and water are principal complete-oxidation products; CO, hydrogen and methane also participate in the finite-rate network.
Small radical pool	`h`, `o`, `oh`, `ch3`, `hco`	Hydrogen, oxygen, hydroxyl, methyl and formyl radicals carry the fast reaction chains. OH is shown in the profile comparisons.
Hydrogen peroxide chemistry	`ho2`, `h2o2`	Hydroperoxyl radical and hydrogen peroxide connect radical storage/production to ignition.
Small oxygenated intermediates	`ch2o`, `ch3cho`, `ch3oh`, `ch3coch3`	Formaldehyde, acetaldehyde, methanol and acetone; formaldehyde is a useful plotted marker of the prepared and reacting kernel.
Heptyl radicals	`c7h15-1` through `c7h15-4`	Fuel-derived radicals with different radical positions, formed after hydrogen abstraction.
Peroxy radicals and hydroperoxides	`c7h15o2-1`, `c7h15o2h-1` and their listed isomers	Oxygen addition creates peroxy chemistry; radicals and stable hydroperoxides are distinct species.
Hydroperoxyalkyl pathway	`c7h14ooh1-3`, `c7h14ooh1-3o2` and their listed isomers	Commonly called QOOH and oxygen-added QOOH families; routes toward low-temperature chain branching.
Ketohydroperoxides	`nc7ket13`, `nc7ket23` and the other `nc7ket*` entries	Intermediates whose formation/decomposition can produce OH and promote branching.
Competing fragmentation/oxygenate pathways	`c7h14-1`, `c7h14o1-3`, plus smaller C2–C6 species	Olefins, cyclic ethers and fragments connect alternative propagation pathways to the small-molecule network.

The pathway roles follow the parent mechanism’s reaction-class discussion; the competition between branching and alternative pathways motivates the NTC interpretation, but the current CFD plots are not a reaction-path flux or sensitivity analysis. Curran et al., reaction classes and low-temperature branching.

Exact inventory used here: Cantera 3.2.0 loads 160 species, 1540 irreversible reaction entries, and zero entries marked reversible from the run’s YAML. These are directed kinetic entries; do not report them as 1540 reversible reaction pairs. The historical LLNL description counts 770 reversible reactions and names its reduction “159 species”; the local filename retains that historical name, but the loaded phase count is authoritative. The mechanism includes elements C, H, O and N. Since n2 is its only nitrogen-bearing species, it cannot predict NO/NO2 chemistry. It is also not a liquid-evaporation, droplet or soot model.

The machine-readable inventory records every exact name, elemental composition and one-based phase position, with mechanism SHA-256 67c3c3b79b328f1942294336a8798d047b51407cac698e7649ed6fc4344f1cef. The complete species list is below, in YAML phase order. Names distinguish positional/electronic isomers and must not be merged merely because their elemental formulas coincide; for example, ch2 and ch2(s) are separate entries. The final absorbed (dependent) closure species is c5h9o1-4.

Complete species list

YAML positions	Exact species names
1–8	`n2`, `ch3`, `h`, `ch4`, `h2`, `oh`, `h2o`, `o`
9–16	`c2h6`, `c2h5`, `hco`, `co`, `co2`, `o2`, `h2o2`, `ho2`
17–24	`c2h4`, `ch3oh`, `ch2oh`, `ch3o`, `ch2o`, `c2h2`, `c2h3`, `c2h`
25–32	`hcco`, `ch2`, `ch`, `ch2co`, `ch2(s)`, `pc2h4oh`, `ch3co`, `ch3cho`
33–40	`c3h5-s`, `c3h4-p`, `c3h5-a`, `c3h6`, `c3h4-a`, `ch3chco`, `c3h5-t`, `c4h6`
41–48	`nc3h7`, `ic3h7`, `c3h8`, `c5h9`, `c4h7`, `c4h8-1`, `sc4h9`, `pc4h9`
49–56	`ch3coch3`, `ch3coch2`, `c2h5co`, `c2h5cho`, `c5h10-1`, `ch2cho`, `c5h11-1`, `c5h11-2`
57–64	`c2h5o`, `c2h5o2`, `ch3o2`, `ch3o2h`, `c3h2`, `o2c2h4oh`, `c2h4o2h`, `c2h3co`
65–72	`c2h3cho`, `c3h5o`, `c3h6o1-2`, `c3h6ooh1-2`, `c3h6ooh2-1`, `nc3h7o`, `ic3h7o`, `nc3h7o2`
73–80	`ic3h7o2`, `c4h7o`, `c4h8ooh1-3o2`, `c4h8ooh1-3`, `nc4ket13`, `c4h8ooh1-2`, `c4h8o1-3`, `pc4h9o2`
81–88	`c3h3`, `hocho`, `c2h3o1,2`, `nc3h7cho`, `nc3h7co`, `c3h6cho-2`, `ch2ch2coch3`, `c2h5coch2`
89–96	`c2h5coc2h4p`, `nc3h7coch2`, `nc4h9cho`, `nc4h9co`, `hoch2o`, `c6h13-1`, `c6h12-1`, `c6h11`
97–104	`nc7h16`, `c7h15-1`, `c7h15-2`, `c7h15-3`, `c7h15-4`, `c7h15o2-1`, `c7h15o2h-1`, `c7h15o2-2`
105–112	`c7h15o2h-2`, `c7h15o2-3`, `c7h15o2h-3`, `c7h14-1`, `c7h14-2`, `c7h14-3`, `c7h13`, `c7h15o2-4`
113–120	`c7h15o-1`, `c7h15o-2`, `c7h15o-3`, `c7h14ooh1-2`, `c7h14ooh1-3`, `c7h14ooh1-4`, `c7h14ooh2-3`, `c7h14ooh2-4`
121–128	`c7h14ooh2-5`, `c7h14ooh3-1`, `c7h14ooh3-2`, `c7h14ooh3-4`, `c7h14ooh3-5`, `c7h14ooh3-6`, `c7h14ooh4-2`, `c7h14ooh4-3`
129–136	`c7h14o1-3`, `c7h14o1-4`, `c7h14o2-4`, `c7h14o2-5`, `c7h14o3-5`, `c7h14ooh1-3o2`, `c7h14ooh2-3o2`, `c7h14ooh2-4o2`
137–144	`c7h14ooh2-5o2`, `c7h14ooh3-1o2`, `c7h14ooh3-2o2`, `c7h14ooh3-4o2`, `c7h14ooh3-5o2`, `c7h14ooh3-6o2`, `c7h14ooh4-2o2`, `c7h14ooh4-3o2`
145–152	`nc7ket13`, `nc7ket23`, `nc7ket24`, `nc7ket25`, `nc7ket31`, `nc7ket32`, `nc7ket34`, `nc7ket35`
153–160	`nc7ket36`, `nc7ket42`, `nc7ket43`, `nc4h9coch2`, `c4h7ooh1-4`, `c5h9ooh1-4`, `c4h7o1-4`, `c5h9o1-4`

NTC behavior

NTC means negative temperature coefficient. Over an intermediate initial temperature interval, increasing temperature reduces effective low-temperature reactivity and increases ignition delay. Oxygen-addition/isomerization routes toward chain branching compete with decomposition and propagation routes; their changing balance can make ignition delay non-monotonic even though individual elementary rates accelerate. This concerns induction chemistry, not a negative temperature response of the already burned gas. Curran et al., n-heptane oxidation.

The existing Cantera scan, using this exact YAML and adiabatic constant-volume reactors, demonstrates at 20 atm a main-ignition delay increase from 0.977 ms at 830 K to 2.489 ms at 960 K. Ignition is defined there by peak $\mathrm dT/\mathrm dt$, not the CFD kernel’s 2000 K crossing. The two-scale CFD starts after chemical preconditioning: its short hot continuation is not by itself a fresh demonstration of NTC.

Homogeneous ignition-delay curves. The ascending portions demonstrate the NTC interval in the exact study mechanism; pressure shifts the interval and delay. This is a mechanism/design test, not a CFD accuracy reference.

Low-temperature and main ignition. The separated heating events explain why a single fixed physical step can face slow induction and a rapid runaway. Each heat-release trace is normalized separately; amplitudes are not comparable.

The earlier hot n-heptane posterior matrix had 20 completed rows, but no common finely resolved truth and different coarse-step end times. It demonstrated cost/robustness rather than a clear accuracy win: at code $1.6\times10^{-3}$, coupled and broadA capped 3/7 physical steps, while Strang and BA-CE6 converged 7/7. Typical large-step amortized cost was about 2 s per printed inner iteration for coupled/broadA versus 0.9–1.0 s for Strang/BA-CE6 on 64 ranks. Those costs are configuration-specific.

Ignition plus front: physical preparation and reference trajectory

The goal is to put a narrow diffusion-active reacting interface and a broader chemically sensitive ignition kernel in the same one-dimensional domain. Use a 10 mm closed tube, 1200 cells ($8.333\,\mu\mathrm m$), periodic transverse width $25\,\mu\mathrm m$, initially 20 atm and zero velocity. Streamwise walls reflect waves; boundary interaction is intentional. This is not a free-flame speed problem and the seeded interface is not an exact steady flame.

The fresh stoichiometric mixture before preparation is

$$ Y_{\mathrm{N_2}}=0.7192878080,\quad Y_{\mathrm{O_2}}=0.2185055960,\quad Y_{\mathrm{nC_7H_{16}}}=0.0622065960. $$

Its preparation-temperature profile is

$$ T_{\rm prep}(x)=750\,\mathrm K+80\,\mathrm K \exp\left[-\left(\frac{x-7\,\mathrm{mm}}{1.8\,\mathrm{mm}}\right)^2\right]. $$

Evolve those parcels isobarically for 1.3273 ms. Separately equilibrate the 750 K fresh mixture at fixed enthalpy/pressure to make the burned seed. Blend prepared and burned compositions and enthalpy, using a smooth transition:

$$ \xi=\operatorname{clip}\left(\frac{x-2\,\mathrm{mm}}{0.2\,\mathrm{mm}}+\frac12,0,1\right), \quad w=\xi^3(10-15\xi+6\xi^2), $$$$ Y(x)=(1-w)Y_b+wY_a(x),\qquad \mathfrak h(x)=(1-w)\mathfrak h_b+w\mathfrak h_a(x). $$

Here $a,b$ mean aged and burned, and $\mathfrak h$ is mixture specific enthalpy, not the pressure sensor. Recover temperature from $(\mathfrak h,p,Y)$ with Cantera; interpolate the common $T/Y$ table in the solver and derive density and conservative energy through its EOS. No arbitrary radical mass fractions are inserted. The actual initial kernel maximum is 1278.41 K, while the burned seed is about 2605.52 K; 750–830 K is the preparation range, not the final CFD initial-temperature range.

Full Strang/ESDIRK2 completes a 0–25 μs pilot and a verified restart continuation to 40 μs, with code ceiling $10^{-4}$ (0.263852 μs). The first run has 95/95 converged physical steps; the continuation adds 62/62. The comparison uses P1 FV, corrected reactive transport, rhsFPPMode=2, 10–400 inner iterations, pseudo-CFL ramp 1→100 in ten iterations and density stopping ratio $10^{-3}$. The combined profile history removes the overlapped restart tail.

Small-step full-Strang trajectory through 40 μs. Temperature, pressure, density and velocity separate the reacting interface near 2 mm from kernel runaway near 7 mm. Expansion and pressure waves show this is not a homogeneous constant-volume reactor. These profiles are a practical baseline, not exact truth.

Species evolution on the same slices. Oxygen consumption, formaldehyde loss, evolving CO and increased OH identify the main kernel reaction. Most original fuel was already transformed by preparation, so fuel alone is a poor runaway marker.

Cell-mean budget estimates near the runaway. Molecular diffusion matters locally at the narrow interface; chemical heating and flow compression/expansion dominate much of the kernel. These offline estimates motivate regional splitting, but do not prove causality by removing transport from the CFD.

Short-window comparison: interface and ignition errors

Every comparison starts from the same saved step-90 state at 23.74670185 μs, ends at 32 μs, and uses serial 64-rank execution on cs. The provisional reference advances that restart with Strang at code $5\times10^{-5}$ (0.131926 μs), 63 physical steps and 126 converged stages. It refines only the short window; it does not remove error accumulated before the common restart or establish spatial convergence.

Define interface $I=[1,3)$ mm (240 cells) and ignition kernel $K=[3,10]$ mm (840 cells). On this uniform grid, for either region $R$,

$$ E_{\rm mean}^{R}=\frac1{N_R}\sum_{i\in R}|T_i-T_{i,\rm ref}|,\qquad E_{\rm max}^{R}=\max_{i\in R}|T_i-T_{i,\rm ref}|. $$

Errors are endpoint differences in kelvin at fixed spatial cells. No front alignment is applied; displacement of a steep interface contributes to error. The kernel includes its cooler shoulders, not only the hottest reacting cells. Mean error is not signed bias, and a small mean can hide a large local maximum.

What does zero error mean? In these temperature bar charts, the zero is defined by the Strang $\Delta t_{\rm code}=5\times10^{-5}$ endpoint profile, not by an exact solution or Cantera reactor. Thus $\Delta T_i=T_i-T_{i,\rm ref}$. An exactly zero mean or maximum absolute difference means every sampled temperature in that region matches the reference at 32 μs; positive and negative errors cannot cancel. It says nothing by itself about other fields, earlier times, or agreement with the continuum solution. The reference has zero error against itself by construction. Rounded displayed zeros need not mean bitwise identity.

What is the dotted “reference refinement” line? It is the same regional mean or maximum absolute temperature difference, but between the two Strang controls at code steps $10^{-4}$ and $5\times10^{-5}$. Both start from the same checkpoint, on the same mesh, and end at 32 μs. This measures sensitivity to halving the reference timestep. It is not a fourth/fifth method’s result, not a convergence tolerance, and not the unknown error of the finer run. The plot legend now explicitly names the two Strang steps.

Dotted-line quantity	Interface [K]	Ignition kernel [K]
Mean absolute Strang step-halving difference	7.66	1.52
Maximum absolute Strang step-halving difference	183.29	30.12

A bar below the dotted line is closer to the fine reference than the coarser Strang control is; it is not thereby proven more physically accurate. Two resolutions alone neither establish an asymptotic order nor provide a rigorous error bound. No Richardson correction or extrapolated truth is used here.

Reference sensitivity at 32 μs. Strang at $10^{-4}$ versus $5\times10^{-5}$ differs by interface mean/max 7.66/183.29 K, and kernel mean/max 1.52/30.12 K. This is an empirical refinement difference, not a rigorous error bound, and prevents overinterpreting small improvements.

Matched actual code step $4\times10^{-4}$

All four methods use seven full steps of 1.055409 μs and a common final shortened step: eight physical steps and sixteen implicit stages. Physical CFL 500 does not clip their histories. The earlier CFL-200 runs with the same nominal ceiling had different actual steps and are kept separately in the study report, not mixed into this table.

Treatment	Interface mean / max [K]	Kernel mean / max [K]	Stages converged / capped	Wall [min]
Coupled	18.04 / 293.91	44.13 / 684.48	16 / 0	16.34
Strang	19.43 / 575.75	11.60 / 346.22	16 / 0	7.69
broadA	18.45 / 365.36	14.20 / 298.20	14 / 2	38.61
Balance	16.85 / 378.28	10.54 / 251.70	16 / 0	8.96

The broadA endpoint is shown diagnostically: one physical step contains two capped stages, so its errors are not accepted as fully converged evidence. Balance has no snapped fully coupled cells in the last frozen field: 2.0% mixed, 98.0% Strang, mean $1-\chi=0.00701$. broadA has 1.33% coupled, 16.08% mixed and 82.58% Strang, mean $1-\chi=0.06951$. These are whole-domain cell fractions, not fractions of heat release or runtime.

The key regional error comparison. Hatched broadA bars denote capped stages. Dotted levels are the separate reference-refinement differences. Here $\Delta T=T-T_{\rm ref}$; the zero baseline is agreement with the fine Strang endpoint, not exact physical truth. Balance reduces the kernel mean error by about 76% and its maximum by about 63% relative to coupled, but coupled has the smallest interface maximum.

Temperature and fuel at the matched endpoint. The interface and kernel are separately enlarged; the fine Strang curve is the provisional reference. Localized peak errors arise near steep fronts and kernel shoulders.

Signed spatial error, not only summary norms. Coupled’s kernel mismatch is broad enough to increase the mean strongly; interface shifts give narrow positive/negative error structures. A better mean does not imply every cell is closer to the reference.

Kernel maximum through the restart window. The dotted 2000 K line is a convenient spatial marker, not the homogeneous peak-heating ignition definition. Crossings are interpolated between saved steps and inherit cadence uncertainty.

Final temperature and last frozen selector. Balance restricts mixing much more than broadA. The selector was evaluated at the start of the final shortened step; it is not a new a-priori evaluation of the endpoint at the full nominal step.

At $2\times10^{-4}$, all three original treatments converge: coupled, Strang and broadA have interface means 18.40, 17.21, 23.29 K and kernel means 9.15, 4.12, 3.56 K. BroadA is therefore not uniformly better as the step decreases. Balance has not been run at that smaller step.

Why these results motivate Balance, but do not finish validation

The strongest present result is a regional tradeoff: coupled has much worse kernel accuracy at matched large step, while Strang has a worse interface maximum. Balance reduces both kernel norms relative to coupled and both interface norms relative to Strang, with every stage converged. Its runtime is about 17% above Strang and 45% below coupled in this one comparison.

However, Balance’s mean-error gains over Strang are only 2.58 K at the interface and 1.06 K in the kernel, smaller than the corresponding reference refinement differences. Coupled still wins interface maximum error. The candidate and older controls also use different executable hashes, although the isolated patch adds mode 3 without changing old modes; no repeated-run timing distribution is available. This is a promising matched-step result, not a universal or reference-independent superiority claim.

Reserved $8\times10^{-4}$ results

All numerical cells below remain placeholders until terminal reduction, restart identity, actual timestep histories and stage convergence are checked. The intended step is 2.110818 μs with physical CFL 1000; four steps would reach the endpoint if unclipped. No half-step smoke result substitutes for this test.

Treatment	Numerical assessment	Interface mean / max	Kernel mean / max	Convergence / wall
Coupled	Failure observed; terminal reduction pending	—	—	—
Strang	Pending integration	—	—	—
broadA	Pending integration	—	—	—
Balance	Pending integration	—	—	—

Execution-only update, 03:47 UTC 29 September: the monitor observed coupled exit 134 after about 52.9 minutes, following a Cantera UV-state-recovery error and ChemicalSource.cpp:757 assertion. The serial launcher advanced to Strang. No completed coupled endpoint or accuracy number is inferred. This note is not a continuously refreshed queue dashboard; the three remaining rows retain placeholders in this curated edition even if their live status later changes.

Reserved figures: matched endpoint profiles, interface/kernel mean and maximum errors, actual timestep history, and convergence/runtime at $8\times10^{-4}$. No empty axes or fabricated zero-error curves are drawn.

What should be tested next

Complete the already-authorized larger-step comparison and retain failed or capped rows visibly. Do not optimize the selector against failed outputs.
Refine the common-window reference once more, preferably with a second converged temporal treatment, before claiming small Balance-vs-Strang wins.
Replay Balance on the H2/O2 flame/detonation snapshot controls before any posterior extension. It must not trade away broadA’s useful flame band.
Check threshold robustness around the selected nice values, with fixed masks and metrics. A narrow “winning” coefficient is not sufficient.
Keep equal-step accuracy primary. Timing supports a possible benefit here, but chemistry kernels, load and inner convergence still confound broad efficiency claims. No extra run is authorized by this reporting list.

Provenance and local reproduction

The study manifest and frozen curation evidence record source hashes, all compact inputs, regional metrics, flame coverage and the 52-row detonation table. Base solver commit: b7cb11e7d53b24346c63128a1a2db335b4d0ac04. Balance uses the isolated codex/rri-diffusion-balance worktree with a patch that was uncommitted at build time, SHA-256 ed2e81a38584b33c672a15126b41ac40217f8953df37b7ee8b86000db18dd614 and executable SHA-256 d02b034370e55c33ca4fe31e1015d5f1b82006418a452c2c713a081dc6362eda. It must not be described as a clean committed build. The original controls use executable SHA-256 198979b0ea9b8cb8ab0f5918ad7da04d0a486bc4035da998beec915aad9f0f5e.

The Balance source is now published as DNDSR commit c8de3c508f87c7b67d05bfff99d6809519a084a8 on codex/rri-diffusion-balance. The commit hook changed only whitespace relative to the recorded run patch. The original patch and build hashes remain intact: publication did not rebuild or replace the running executable.

Run from this repository with NumPy and Matplotlib installed:

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python studies/nheptane_two_scale_strang/scripts/curate_reports.py
python studies/nheptane_two_scale_strang/scripts/plot_candidate.py
python studies/nheptane_two_scale_strang/scripts/plot_comparison.py
python studies/nheptane_two_scale_strang/scripts/analyze_balance.py

The first command regenerates dated-summary PNG/PDF figures and evidence from compact arrays, excluding all $8\times10^{-4}$ posterior rows by design. It does not rewrite the narrative reports or start a solver. The other scripts reproduce the trajectory, earlier comparisons and frozen selectors. The NTC/flame figures retain their source-study reproduction instructions. No raw VTU/VTKHDF or remote access is needed to regenerate the illustrated profiles.