CFD 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.

Orion with LVR: steady robustness, IDDES trials and an SA-R correction

Mon, 19 Oct 2026 13:37:23 +0800

Recorded October 19, 2026. This report consolidates the Orion experiments on thtj1 and the separate NACA0012 source-model check. The main outcome is a useful steady robustness result for LVR v3, rather than a demonstrated high-order DES result. P3 LVR retains plausible fields through 20,000 steady updates; full P3 3WBAP becomes unphysical. Neither reaches the requested steady density threshold. O2 WBAP IDDES advances reliably with converged implicit stages, but the high-order LVR IDDES attempts do not yet provide an acceptable transient result.

All Orion results discussed here were produced with the reversed, historical version-0 SA rotation correction. The corrected source now defaults to standard SA-R, and SAVersion=9001 preserves the old behavior for reproduction. That code repair does not change any archived Orion result. The O2 IDDES case is explicitly an O2 control, not an LVR run.

The source repair and historical selector are committed on DNDSR dev/harry as 03891f8b. Both SA solver variants build, the two SA schemas were regenerated, and the focused regression passes 30 cases / 164 assertions. This is focused validation rather than a new full CPU-suite run.

1. Geometry, physical conditions and checkpoint

Quantity	Setting
Geometry	Orion capsule half domain, symmetry plane $z=0$; redraw mesh, 8,857,647 cells
Mesh	`Orion_str.cgns-merged-redraw3-A1.cgns`, existing 896-partition mesh input
Freestream	Mach 6.41, angle of attack $28^\circ$, zero sideslip, $Re_D=1.077\times10^7$
Thermodynamics	$T_\infty=73.63$ K, $p_\infty=4068.260149$ Pa, isothermal wall at 295.56 K
Scaling	Diameter $D=0.254$ m = 10 solver length units; $U_\infty=1102.53$ m/s = 1 solver velocity unit
Boundary conditions	Original inflow/farfield, wall and symmetry setup; the back face uses the archived inflow condition, termed backin
Initial state	Original O2 Roe-M8 backin steady checkpoint at internal update 20,000, not a failed O4 checkpoint
Resources	thtj1, cp6, 16 nodes × 56 ranks/node = 896 MPI ranks; one thread/rank; 24-hour allocations
Runtime	Open MPI 4.1.4; source/executable/config hashes retained in the per-run manifests

The starting checkpoint is out-restart1-restart1-rot-rm8-cw-backin__1_20000_p896.restart.dnds.h5, SHA256 3935c12bcc05caeec2841ffb7843684ee11b0665ce0351d3f7283b8e38bb2b07. It is a partitioned HDF5 DNDS restart. The checkpoint audit identified it as a practical O2 warm start with stable late forces, not a residual-converged reference solution. The historical HLLEP/farfield and Roe/backin runs have different flux/boundary settings and cannot serve as a matched limiter comparison.

The subsequent steady tests change to Roe M9, rotated scheme 1 with Roe M2/Rusanov in the auxiliary direction. They preserve QCR, enabled SA rotation correction, second-order SA reconstruction (ransForce2nd=1), wall treatment and the original O2 restart. The boundary-cell contours below use jet color scales and cell averages without cell-to-point interpolation. Axes are normalized by body diameter.

2. What LVR means in this test

The high-order fluid reconstruction is a degree-three polynomial, P3, formally fourth order in smooth flow, with HQM_OPT derivative weights and integration order 5. The explicit GG O2 reference uses characteristic WBAP limiting and facial positivity protection. Both the shock detector and the penalty target use that limited reference.

The selected method is LVR v3 = Average H2, with separate shock and contact caps $(\alpha_{s,\max},\alpha_{\rho,\max})=(1,1)$ and the weak-shock extension disabled. The production input uses pressure thresholds $(j_{p0},j_{p1})=(0.04,0.16)$ and compression thresholds $(j_{c0},j_{c1})=(0.03,0.13)$. The contact face-density ramp is $(0.5,0.8)$, supplemented by cell-mean-supported density-extremum treatment with two neighbour layers; the full merged input records the other support thresholds. A cap of one permits complete recovery of the limited O2 target where the gate saturates. It does not impose O2 reconstruction throughout the domain.

For orientation, let $a_i$ be the vector of high-order coefficients in cell $i$, $a_i^{(2)}$ the embedded limited O2 target and $A_i$ the original VR diagonal block. A frozen-gate reconstruction equation has the form

$$A_i a_i+\sum_{j\ne i}A_{ij}a_j=b_i+C_i(a_i^{(2)}-a_i).$$

Here $b_i$ is the original cell-mean-dependent VR right-hand side, $A_{ij}$ are neighbour couplings and $C_i$ is the nonnegative penalty block. With $C_i=\lambda_i A_i$ and $\alpha_i=\lambda_i/(1+\lambda_i)$, a block-Jacobi or block-GS update is

$$a_i^{\rm new}=(1-\alpha_i)A_i^{-1}\left(b_i-\sum_{j\ne i}A_{ij}a_j^*\right)+\alpha_i a_i^{(2)}.$$

The neighbour values $a_j^*$ come from the previous sweep for Jacobi, and from the most recent available values for GS. This reuses the original diagonal inverse. The gate and target depend nonlinearly on cell means, while the frozen reconstruction solve remains linear in $a_i$. Zero gate leaves VR unchanged; a saturated gate returns the O2 target. This is an explanation of the reconstruction mechanism, not a claim that the gated transient CFD system is linear or automatically positivity preserving. The case retains its separate positivity safeguards.

See the steady setup and effective controls for the exact production configuration.

3. Steady P3 LVR versus full P3 3WBAP

The two terminal controls share the original O2 checkpoint and steady controls: pseudo-time CFL 2, four local time-step smoothing passes, density-relative threshold $10^{-5}$ and cap 20,000 updates. Their printed CFL schedule starts and ends at 2, so it supplies no actual CFL increase. The conventional P3 3WBAP control is fully applied, not selectively restricted by a shock detector.

Method	Job	Terminal update	Absolute density $L_1$ residual	Relative density	$C_L$	$C_D$	Assessment
LVR v3, WBAP O2 target, HQM_OPT, (1,1)	12060729	20,000	17.0713	0.0334118	-0.236226	0.559615	Plausible saved state; density unconverged
Full 3WBAP, P3, HQM_OPT	12051855	20,000	7,688.68	0.0263359	34,852.2	-64,787.8	Unphysical numerical result

Residuals are the absolute volume-weighted $L_1$ values from the solver .log CSV. Relative density uses the running-maximum reference of each solve, so that column is useful for the stopping criterion but not for ranking the two different trajectories. The figures compare absolute residuals. Following the researcher’s convention, SA is a stability diagnostic, not an additional stopping requirement.

LVR’s last-500 lift range is $[-0.236229505,-0.236226445]$ and drag range $[0.559604771,0.559614566]$. These stable force tails coexist with a nonzero density residual and must not be described as a converged steady solution. The separated wake is expected to contain unsteady dynamics; this cap-limited steady solve is mainly a robustness check and possible starting state.

Both frames use Mach 0–7, temperature 0–800 K, pressure ratio $p/p_\infty$ 0–55 and density ratio 0–14. The LVR symmetry plane spans approximately 62.67–680.53 K and pressure ratio 0.166–52.80. The full-3WBAP plane spans 5,156.75–668,638.9 K, pressure ratio 7.37–808,216 and density up to 677.79. Its saturated colors expose an invalid state rather than an unusually strong physical shock. Finite positive fields and a zero process exit are insufficient acceptance tests.

The practical finding is that this LVR setup preserves a useful high-order steady iterate on this difficult case, while this fully limited P3 3WBAP setup fails numerically. It does not establish universal superiority over conventional limiters, and the SA source error is shared by both methods.

Cost and cluster effects

Run	Scheduler result	Accounted elapsed	Suspended time	Note
Steady LVR 12060729	COMPLETED, exit 0:0	12:51:25	0	Reached update cap, not density criterion
Full 3WBAP 12051855	COMPLETED, exit 0:0	15:21:54	02:21:07	Numerically invalid despite normal exit
O2 IDDES 12056001	TIMEOUT	24:00:17	02:21:06	Allocation ended during physical step 5426

The late-500 median printed update times are 2.27031 s for LVR and 2.75035 s for 3WBAP. The corresponding reconstruction/RHS/communication times are 0.12217/1.14663/0.34882 s and 0.11395/1.09973/0.38918 s. The conventional limiter timer is about 0.71949 s for 3WBAP, versus approximately zero for the LVR post-limiter slot; LVR still has reference/gate work, including limiter-B time around 0.31296 s. Printed timers may overlap and are not exclusive additive components. These late timings compare different numerical states, so they are descriptive rather than a controlled cost-to-accuracy benchmark.

Earlier steady runs suffered an MPI daemon communication failure. A later LVR allocation was anomalously slow; a matched fresh allocation improved measured startup throughput by 7.85 times without changing numerical controls. These incidents are retained in the failure/restart history and allocation replacement record. Scheduler suspension also lengthens calendar intervals beyond accounted active time. Numerical breakdown and cluster transport/performance incidents are different failure modes.

4. Time-accurate work: high-order limitation and O2 control

LVR IDDES attempts

The first P3 LVR v3 (1,1) IDDES run used ESDIRK3, step 0.02 and a 100-update inner cap. It failed numerically during physical step 5, stage 3, at time 0.08. Earlier implicit stages had hit the cap. A retry with maximum step 0.005 and the existing positivity-controlled time-step selector reduced the selected step from $4.40\times10^{-4}$ on step 1 to $9.46\times10^{-9}$ on step 2; it did not demonstrate useful physical-time progress. The retry was cancelled and retained. These are failed or impractical high-order trials, not valid DES results.

The production control therefore returned to explicit GG O2 with full characteristic WBAP. It starts from the same archived O2 checkpoint, uses maxOrder=1 and intOrder=1, and retains the same Roe M9/rotated-M2 and SA/QCR choices. It does not activate the LVR penalty. See the IDDES experiment history.

O2 IDDES controls and results

Quantity	Production control
Integrator	ESDIRK3, `odeCode=202`, three implicit stages per physical step
Physical step	Fixed 0.005 solver time units, no PP or CFL physical-step restriction
Inner solve	Density-relative tolerance $10^{-3}$, cap 500 per implicit stage
Pseudo-time CFL	0.5 to 20 over the first 25 ramp intervals, reset for every stage
IDDES	Mode 1, nominal scale 0.65, $f_{t2}$ and low-Re correction $\psi$ enabled
Output	Cell-average VTKHDF and HDF5 restart every 500 physical steps, step named; point output disabled
Target	10,000 physical steps; 24-hour allocation

The 0.5-to-20 schedule was selected from matched three-step tests. Summed inner time fell from 436.424 s for the 0.5-to-2 baseline to 157.154 s, a 2.777-fold short-startup improvement. All nine stages of each tested schedule reached the same density criterion and the final volume checks passed. This is a measured startup result, not an estimate based only on nominal CFL. The full CFL comparison preserves the stage counts and absolute-residual checks.

Job 12056001 completed 5,425 physical steps, reaching time 27.125, before the allocation timed out. All 16,275 completed implicit stages passed the density criterion; none reached the cap. The largest terminal stage iteration was 47. The median over the last 100 complete steps was 7 stage iterations and 13.7255 s per physical step. The last completed forces were $C_L=-0.2365361$ and $C_D=0.5579229$. The final interrupted stage is not counted as converged.

The latest saved frame is step 5,000, time 25, not the final completed step 5,425. Since $D=10$ and $U_\infty=1$ in solver units, it represents 2.5 diameter convection times. Step 0.005 corresponds to approximately 0.11519 microseconds physically.

5. Wake evolution and model-switch diagnosis

For the symmetry-plane window $0.65nuTilde is primitive normalized working viscosity; saved density multiplied by it gives the normalized transported SA state \(q=\rho\widetilde\nu$. Define mean speed $v=|\mathbf u|/U_\infty$ and the field-change measure

$$E_f(t)=\frac{\sqrt{\sum_i w_i[f_i(t)-f_i(0)]^2}}{\sqrt{\sum_i w_i f_i(0)^2}}.$$

Here $f$ is either $q$ or $v$. This is the spatial RMS difference from initialization, not the temporal fluctuation intensity. These are symmetry-plane area statistics, not whole-volume turbulence statistics.

Saved step	Solver time	Mean $q$	$E_q$	Mean $v$	$E_v$
0	0	2.24959	0	0.791041	0
1500	7.5	3.00963	56.94%	0.790592	1.728%
3000	15	4.27622	115.67%	0.789814	2.002%
5000	25	5.68723	170.17%	0.789046	2.721%

Mean transported SA state increases 152.81%, while mean speed changes -0.252%. The primitive working-viscosity mean increases 402.23%; its mean differs from the density-weighted quantity. These observations demonstrate continuing model-state adjustment with comparatively small velocity changes, not a fully developed or statistically stationary DES wake.

The nominal coefficient 0.65 is not a direct cap on $\widetilde\nu$. With physical wall distance $d$ and the solver’s bounding-box cell size $h$, the existing caller supplies

$$\ell_{\rm LES}=\min\{0.65h,\;0.15\max(d,h)\}.$$

The source then blends

$$\ell_{\rm IDDES}=\widetilde f_d(1+f_e)d+(1-\widetilde f_d)\psi\ell_{\rm LES},$$

where $\widetilde f_d$ is the RANS-side shielding weight, $f_e\ge0$ is the IDDES elevating function, and $1\le\psi\le10$ is the low-Re correction. Thus released shielding need not imply a shorter effective length: a large $\psi$ can enlarge the LES contribution. A very large nominal DES scale also does not algebraically bypass every part of the mode-1 blend; the steady cases are nominal RANS-intended controls with this retained implementation caveat.

An offline 3-D sample replay of frames at times 0, 7.5 and 15 estimates that about 98% of retained wake volume is on the LES side of shielding, while only 60–61% has effective length below $0.9d$. The $\psi>9.9$ volume fraction is about 53–57%. These estimates use least-squares gradients from sampled cell means, not the solver’s stored limited gradients. Their shielding classification is consistent between 16- and 32-neighbour fits; thin-wall coverage and quantitative production budgets are inadequate. Mean modeled-viscosity ratio in the retained wake is only about 0.31–0.46, so a high $\psi$ does not establish excessive eddy viscosity everywhere.

See the formulation audit and diagnostic definitions and limitations. Neither the SA growth nor the small speed change can be attributed to one model term using this evidence alone.

6. SA-R correction and the independent NACA0012 check

Let $G_{ij}=\partial u_i/\partial x_j$, $S_{ij}=(G_{ij}+G_{ji})/2$ and $W_{ij}=(G_{ij}-G_{ji})/2$. Strain and vorticity magnitudes are $s=\sqrt{2\sum S_{ij}^2}$ and $\omega=\sqrt{2\sum W_{ij}^2}$. The production term for positive working viscosity is

$$P=c_{b1}(1-f_{t2})(\widehat S+\Delta)\widetilde\nu,$$

where $c_{b1}=0.1355$, $f_{t2}=1.2\exp[-(\widetilde\nu/\nu)^2/2]$, $\nu$ is laminar kinematic viscosity and $\widehat S$ is the existing modified SA vorticity frequency. Standard SA-R uses $\Delta=2\min(0,s-\omega)$, suppressing production in rotation-dominated regions. The archived Orion implementation reverses the subtraction.

Selector in the repaired source	Meaning	Added production frequency $\Delta$
`SAVersion=0`, default	Corrected SA-R	$2\min(0,s-\omega)$
`SAVersion=9001`	Bad historical version 0, for reproduction	$2\min(0,\omega-s)$
`SAVersion=1`	Oldest retained legacy formulation	$\min(0,\sqrt{2}s-\omega)$

Versions 0 and 9001 share the standard strain normalization, coefficient 2 and negative-production contribution in the approximate source diagonal. Version 1 also changes the normalization, uses coefficient 1 and omits production from that diagonal. It cannot isolate the sign correction. The correction acts on production, not on the IDDES blend or LVR gate. No archived Orion run has been rerun with corrected SA-R in this report.

The independent NACA0012 check uses the supplied AOA15 input: Mach 0.15, $Re_c=2.88\times10^6$, AoA $15^\circ$, 20,816 cells, Roe M8, HQM_OPT P3 reconstruction, limiter off, QCR off, common uniform start, pseudo-time CFL 50 and density threshold $10^{-7}$. Each run is serial on 16 local ranks with a 900-second bound. SA stability is checked without requiring its residual to converge.

Source behavior	Updates to density criterion	Wall minutes	$C_L$	$C_D$
Corrected version 0	3,085	11.49	1.472217	0.0286626
Bad version 0	3,341	9.07	1.521210	0.0239430
Legacy version 1	2,999	10.89	1.472307	0.0288519

All three meet the density criterion without blow-up. Corrected 0 and legacy 1 differ by 0.0061% in lift and 0.6604% in drag. Corrected 0 differs from bad 0 by -3.22% in lift and +19.71% in drag. Repeated version-0 trajectories match their earlier trials exactly at saved precision, but machine timings vary substantially; elapsed times do not establish a source-formula speedup. These are model-sensitivity results, not an experimental accuracy ranking.

The NACA runs used preserved corrected/original executables. The new 9001 selector preserves the original executable’s bad-0 formula; it is not a claim that a new 9001 NACA run was performed. Analytic kernel regression covers all three formulas in 2-D/3-D and their implicit diagonals. See the full three-way study.

7. Interpretation and next experiments

The strongest positive evidence is steady robustness of P3 LVR v3 on Orion. Its stable force tail and plausible fields are useful, while the density residual prevents a steady-convergence claim. Fully limited 3WBAP is an unsuccessful control for these settings. The evidence is specific to this mesh, boundary setup, source model and restart.

The main unresolved issue is time-accurate high-order robustness. The successful transient history belongs to O2 WBAP IDDES, and spans only a few diameter convection times. It cannot establish that LVR resolves the unsteady wake, or that the flow has developed DES statistics. Shared SA and length-model qualifications also prevent attributing missing structures solely to reconstruction dissipation.

The next discriminating steps are:

Run matched corrected-SA-R and 9001 controls from the same O2 checkpoint, keeping every other numerical and model choice fixed. This isolates the rotation sign before retuning DES coefficients.
Record exact solver shielding, effective length, $\psi$, modeled viscosity and source budgets. Offline reconstructed gradients are inadequate for thin-wall production balances.
Reintroduce high-order LVR with controlled physical-step and inner-solve comparisons against the accepted O2 transient control. Preserve failed/cap-limited trials as part of the evidence.
Extend accepted transient runs across more body convection times and assess wake spectra, resolved fluctuations and mesh/time sensitivity. A smooth snapshot or stable force trace alone does not establish the quality of DES.

These are proposed follow-ups; no new Orion job was launched for this report.

8. Evidence map

Terminal Orion reduction, all three jobs, timing, history rollover and checksums.
Steady LVR/full-3WBAP setup and chronological records.
IDDES failed attempts, O2 production and CFL tuning.
SA-R correction, analytic regression and original two-way trial.
Three-way 900-second NACA comparison and matched fields.
Report asset provenance and committed-source validation.

The terminal IDDES CSV rolls after 100,000 records; both continuous segments were recovered and reduced. Steady no-update terminal reports and outer-summary rows are excluded from update/force statistics. Exact remote stdout/snapshot locations and SHA256 values are in the retrieval manifest. Raw fields and full stdout remain in the DNDSR data area; compact evidence, scripts and report figures belong to this independent LVR repository.

The named study records above refer to the private LVR research archive. This article includes its numerical conditions, qualification of failed/partial runs and all displayed figures locally. The source change is DNDSR commit 03891f8b; the archived Orion executables predate that correction.

DNDSR Euler3D P3 weak scaling on four CPU clusters

Mon, 19 Oct 2026 11:18:05 +0800

How well does DNDSR scale when each CPU core has roughly the same amount of work? This study measures its fourth-order Euler3D solver on four CPU clusters. The largest accepted production run reaches 8,192 MPI ranks, but the clusters differ sharply in how they behave at scale.

Every run uses whole nodes and one MPI rank per physical core. The weak-scaling runs and workload-matched retests keep roughly 1,024 periodic hexahedral cells per rank; four diagnostic controls use heavier workloads. The measurement covers the steady internal update of the P3 VFV scheme (maxOrder=3, quadrature intOrder=5, limiter disabled), not mesh setup.

Each completed run records 500 solver updates. The first 100 serve as warmup; all 400 remaining updates enter the statistics without trimming. The mean includes slow updates and stalls, while the median better describes a typical update. Their difference matters most at high rank counts, where a few long updates can lower sustained throughput. A cell-iteration is one cell processed in one solver update; MCell-it/s means millions of these per second. Timing CV describes the relative variation among update times.

The audit contains 31 runs with all 400 measured updates:

22 production weak-scaling runs using serial CGNS input;
five workload-matched distributed-HDF5/new-MPI retests;
four distributed-HDF5 diagnostic controls with deliberately heavier per-rank workloads.

Runs that stopped during mesh setup produced no update timings, so they are not performance points.

Machines and MPI stacks

Cluster	CPU per node	Physical cores/node	Benchmark partition	Production MPI	Supplemental MPI
m8	2x AMD EPYC 9654, Zen 4	192	`amd_m8_768`	Open MPI 4.1.0, UCX 1.14	Open MPI 5.0.10, UCX 1.14
m9	2x AMD EPYC 9755, Zen 5	256	`amd_m9_768`	Open MPI 4.1.8, UCX 1.18	same stack
a8	2x AMD EPYC 9554, Zen 4	128	`amd_a8_384`	Open MPI 4.1.0, UCX 1.14	Open MPI 4.1.6, UCX 1.14
thtj1	2x Intel Xeon Gold 6348, Ice Lake-SP	56	`cp6`	Open MPI 4.1.4, site UCX/GLEX	none

These are the normal production partitions: a8 used its 384 GB class rather than the dedicated 768 GB large-memory partition, and m9 used 768 GB rather than 1.5 TB. m8 used its M8-specific 768 GB partition; thtj1 used cp6.

Production weak scaling

Production weak scaling. Throughput continues to rise to m9’s 8,192-rank point, although its efficiency falls. thtj1 shows a different pattern: a sharp throughput drop beyond 16 nodes.

Download the vector PDF.

The horizontal axis counts physical CPU cores. The left panel shows stall-inclusive mean throughput; the right panel compares that throughput with each cluster’s one-node baseline.

Cluster	Nodes	Ranks	Cells/rank	Mean MCell-it/s	Median MCell-it/s	Mean efficiency	Median efficiency	Timing CV
m8	1	192	1016.2	1.772	1.830	100.0%	100.0%	0.108
m8	2	384	1013.1	3.504	3.672	98.9%	100.3%	0.150
m8	4	768	1013.9	6.261	6.527	88.3%	89.2%	0.161
m8	8	1536	1016.2	9.366	12.569	66.1%	85.8%	2.941
m8	16	3072	1034.0	20.358	23.774	71.8%	81.2%	0.307
m9	1	256	1024.0	2.382	2.808	100.0%	100.0%	1.599
m9	2	512	1038.0	5.103	5.450	107.1%	97.1%	0.371
m9	4	1024	1036.3	9.238	10.545	96.9%	93.9%	0.657
m9	8	2048	1024.0	15.264	20.323	80.1%	90.5%	0.773
m9	16	4096	1018.9	21.936	34.915	57.5%	77.7%	1.077
m9	32	8192	1021.2	30.201	44.441	39.6%	49.5%	0.978
a8	1	128	1036.3	1.656	1.712	100.0%	100.0%	0.082
a8	4	512	1038.0	5.249	6.177	79.3%	90.2%	2.092
a8	8	1024	1036.3	11.211	11.790	84.6%	86.1%	0.171
a8	16	2048	1024.0	19.793	21.288	74.7%	77.7%	0.173
thtj1	1	56	1059.3	0.525	0.528	100.0%	100.0%	0.029
thtj1	2	112	1050.4	1.030	1.040	98.0%	98.4%	0.042
thtj1	4	224	1013.3	2.036	2.076	96.9%	98.2%	0.060
thtj1	8	448	1019.0	3.980	4.109	94.7%	97.2%	0.103
thtj1	16	896	1018.6	7.309	7.639	87.0%	90.4%	0.119
thtj1	32	1792	1013.3	2.101	2.109	12.5%	12.5%	0.011
thtj1	64	3584	1019.0	4.414	4.241	13.1%	12.5%	0.177

m9 reaches the largest accepted production point: 8,192 ranks on 32 nodes. It delivers 30.20 million cell-iterations/s by the mean, versus 44.44 million by the median. Mean and median efficiencies are 39.6% and 49.5%. That gap, alongside a timing CV of 0.978, shows how strongly slow updates affect the sustained average.

m8 and a8 retain 71.8% and 74.7% mean efficiency at 16 nodes. thtj1 also scales smoothly to 16 nodes, then drops from 7.31 to 2.10 million cell-iterations/s at 32 nodes. Its 32-node timing CV is only 0.011: the slowdown is consistent, not the result of a few outlier updates. Throughput partly recovers at 64 nodes, but efficiency stays near 13%. The attempted 128-node runs never reached solver updates and therefore have no throughput measurement.

Per-core throughput and MPI retests

Throughput per core. Production and workload-matched HDF5/MPI retest series are distinguished in the legend. Completing a large run does not guarantee strong per-core performance, particularly on m8.

Download the vector PDF.

The five HDF5 retests keep the workload within 1.2% of the 1,024-cell-per-rank target. They use validated MPI stacks available on the same machines.

Cluster	Nodes	Ranks	Cells/rank	MPI stack	Mean MCell-it/s	Median MCell-it/s	Mean kCell-it/s/core	Median kCell-it/s/core	CV	Outcome
a8	1	128	1036.3	Open MPI 4.1.6 + UCX 1.14	1.611	1.679	12.584	13.116	0.153	completed
a8	32	4096	1018.9	Open MPI 4.1.6 + UCX 1.14	29.751	33.806	7.263	8.253	0.561	completed
m8	1	192	1016.2	Open MPI 5.0.10 + UCX 1.14	1.792	1.954	9.335	10.177	0.299	completed
m8	32	6144	1030.5	Open MPI 5.0.10 + UCX 1.14	12.421	13.758	2.022	2.239	0.944	completed
m9	16	4096	1018.9	Open MPI 4.1.8 + UCX 1.18	21.218	27.865	5.180	6.803	0.977	solver completed; allocation cleanup timed out

On m8 and a8, the newer Open MPI stacks complete workloads at scales where the earlier stacks failed during setup. The executable and UCX version were retained; HDF5 input alone did not make the old Open MPI 4.1.0 large-rank attempts complete. These checks support an MPI-stack explanation for the completion difference, but not a general speedup.

Performance remains uneven. Relative to the matching one-node retest, a8 retains 57.7% mean and 62.9% median efficiency at 32 nodes. m8 retains only 21.7% mean and 22.0% median efficiency, with CV 0.944. Its newer stack solves the completion problem, not the large-scale performance problem.

The m9 HDF5 run produced all 400 samples, and the application and MPI steps exited successfully. The outer allocation reached its shortened time limit only during cleanup. Its performance sample is therefore included, with that status noted in the table.

Heavier diagnostic controls

The remaining four completed runs have 1.33 to 15.92 times the target cells per rank. Their timings are valid for those heavier workloads, but they do not belong to the workload-matched weak-scaling curves.

Cluster	Nodes	Ranks	Cells/rank	Relative load	Mean MCell-it/s	Median MCell-it/s	Mean kCell-it/s/core	Median kCell-it/s/core	CV
a8	16	2048	2037.7	1.99x	20.049	21.133	9.789	10.319	0.117
a8	24	3072	1358.5	1.33x	23.292	25.871	7.582	8.422	0.599
m8	16	3072	2061.1	2.01x	21.520	24.011	7.005	7.816	0.196
m9	1	256	16301.9	15.92x	2.716	2.785	10.608	10.879	0.160

Interpretation and scope

The benchmark reveals three distinct behaviors:

m9 runs the unchanged production path at 8,192 ranks, beyond 5,000. There is no universal rank-count ceiling near that value.
On m8 and a8, repeatable large-rank setup failures are associated with the production Open MPI 4.1.0 stacks. Newer validated stacks complete at the same scale, although m8’s sustained efficiency remains poor.
thtj1 has a different limit: its 32-node updates are consistently slow, and its 128-node attempts spend hours in distributed mesh setup without producing update timings.

This is a whole-node, pure-MPI application benchmark, not a controlled comparison of CPU architectures. Core counts, processors, MPI and UCX versions, and network fabrics differ among clusters. Read mean/median gaps and timing CVs alongside throughput rather than judging a cluster by a single high-rank result.

The solver source is DNDSR dev/harry commit f6ece7e91e97c6337ef0761c30017b2ddad6b24b. Exact configuration hashes, machine-readable timing rows, scheduler classifications, and failure evidence are in the public benchmark report.

Limited Variational Reconstruction v2: formulation and validation

Tue, 29 Sep 2026 12:08:19 +0800

Updated 2026-09-29. This post collects the mathematical definition, method-development path, Lax shock-band refinement, and the completed or active production evidence for Limited Variational Reconstruction v2 (LVR v2). Raw solver output remains outside this repository. The linked study records contain the exact configurations, executable and mesh hashes, commands, manifests, and acceptance checks.

Purpose and current status

LVR changes the reconstruction solve itself. It does not first compute an unlimited high-order polynomial and then replace or rescale that polynomial in a separate limiter pass. A shock sensor built from cell averages and an explicit limited second-order reconstruction selects a penalty that pulls the high-order variational reconstruction toward the limited second-order polynomial. The reconstruction system remains linear in the high-order coefficients during each solve, while its target and penalty strength depend nonlinearly on the current cell averages.

The current candidate is LVR v2:

P3 variational reconstruction;
a characteristic WBAP-limited explicit Green–Gauss O2 polynomial used for both sensing and the penalty target;
the pressure-jump/compression product gate;
a shock-supported pressure-extremum correction that closes the weak-gate shoulder beside a strong one-dimensional shock;
conditional retention and a bounded, one-neighbour seed extension;
positivity protection of the O2 facial states;
HQM_OPT derivative weights as the primary candidate and Factorial weights as a maintained comparison;
maximum penalty fraction $\alpha_{max}=0.5$ in the canonical v2 setting.

The internal configuration value gateMode=8 selects this implementation. It is an enum value, not an “eighth formula”; the public method name is LVR v2.

Evidence set	Current state
Lax shock-band reduction	complete; near-shock plateau oscillation reduced by about 94%
Extended one-dimensional and cylinder validation	complete, with the recorded Roe double-rarefaction limitation
Canonical 13-method compact matrix	all 78 runs complete
Canonical 13-method DM240U matrix on `thtj1`	all 13 accepted runs complete
Structured 400 x 400 2-D Riemann on `thtj1`	all four runs complete
Delaunay 246U 2-D Riemann on `thtj1`	all four accepted runs and final snapshots complete

Terminology used in this post

Term	Meaning in these studies
cell mean or cell average	finite-volume conserved state stored for one cell
O2 or Explicit GG O2	linear, formally second-order polynomial built by an explicit Green–Gauss gradient
P3	degree-three polynomial reconstruction, formally fourth order in a smooth region
VR	the unpenalized variational reconstruction solve
LVR	VR with the shock-dependent penalty toward a limited O2 target
LVR v2	the selected product gate plus shock-supported pressure-extremum correction defined below
WBAP	the characteristic face-based weighted biased averaging limiter used for the selected O2 target
Barth	the scalar Barth limiter available as an alternative O2 target
3WBAP	the code’s conventional fully limited P3 WBAP path
CWBAP	the code’s conventional compact WBAP variant for fully limiting P3
HQM_OPT, Factorial	two derivative-weight schemes used to define the P3 variational-reconstruction functional
Rusanov or Roe M2	local Lax–Friedrichs/Rusanov numerical flux in the DNDSR configuration
Roe M8/M9	entropy-fixed, H-corrected Roe-family flux variants tested in the extended studies
$\rho$	density
$\rho E$	total energy per unit volume
$\chi$	normalized LVR gate $\alpha/\alpha_{max}$ used in contour plots

Reconstruction penalty: self-contained definition

Consider one finite-volume cell $i$. Its cell mean is fixed during a reconstruction solve. The unknown $\mathbf r_i$ contains all non-mean polynomial coefficients for every reconstructed physical variable in that cell. Let $\mathcal N(i)$ be the cells sharing an internal face with cell $i$. The ordinary linear variational-reconstruction block equation is

$$ \mathbf A_i\mathbf r_i =\sum_{j\in\mathcal N(i)}\mathbf B_{ij}\mathbf r_j+\mathbf b_i. $$

Symbol	Definition
$i$	cell whose reconstruction is being updated
$j$	face-neighbour cell of $i$
$\mathcal N(i)$	set of cells sharing an internal face with $i$
$\mathbf r_i$	vector of high-order polynomial coefficients in cell $i$, excluding the fixed mean
$\mathbf A_i$	cell-local diagonal block of the variational-reconstruction system
$\mathbf B_{ij}$	block coupling neighbour $j$ into cell $i$
$\mathbf b_i$	terms fixed during this reconstruction solve, including dependence on current cell means

The unpenalized block Jacobi or block Gauss–Seidel target is

$$ \mathbf G_i=\mathbf A_i^{-1} \left(\sum_{j\in\mathcal N(i)}\mathbf B_{ij}\mathbf r_j+\mathbf b_i\right). $$

Here $\mathbf G_i$ means the update the original VR scheme would produce before relaxation. Let $\mathbf r_i^{O2}$ be the coefficient vector of the explicit, limited O2 polynomial. LVR adds the local quadratic penalty

$$ J_i^{LVR}=\frac{\lambda_i}{2} (\mathbf r_i-\mathbf r_i^{O2})^T\mathbf A_i (\mathbf r_i-\mathbf r_i^{O2}), $$

where $\lambda_i\ge0$ is the unbounded penalty strength and the superscript $T$ denotes transpose. Choosing the penalty matrix as $\lambda_i\mathbf A_i$ is the key algebraic step: it preserves the cached local inverse structure. The penalized cell equation becomes

$$ (1+\lambda_i)\mathbf A_i\mathbf r_i =\sum_{j\in\mathcal N(i)}\mathbf B_{ij}\mathbf r_j +\mathbf b_i+\lambda_i\mathbf A_i\mathbf r_i^{O2}. $$

Define the bounded penalty fraction

$$ \alpha_i=\frac{\lambda_i}{1+\lambda_i}, \qquad 0\le\alpha_i\le1. $$

The exact penalized block target and relaxed iteration are then

$$ \mathbf T_i=(1-\alpha_i)\mathbf G_i+\alpha_i\mathbf r_i^{O2}, $$$$ \mathbf r_i^{new} =(1-\omega_i)\mathbf r_i^{old}+\omega_i\mathbf T_i. $$

Symbol	Definition
$J_i^{LVR}$	quadratic penalty added in cell $i$
$\lambda_i$	unbounded penalty strength; $0$ gives ordinary VR and $\lambda_i\rightarrow\infty$ gives the O2 target
$\alpha_i$	bounded penalty fraction corresponding to $\lambda_i$
$\mathbf r_i^{O2}$	limited explicit O2 target polynomial in cell $i$
$\mathbf T_i$	exact local target of the penalized block equation
$\omega_i$	existing block Jacobi/GS/SOR relaxation factor
$\mathbf r_i^{old},\mathbf r_i^{new}$	coefficient vectors before and after one reconstruction iteration

Thus $\alpha_i=0$ exactly recovers the original VR update. The limit $\alpha_i=1$ selects $\mathbf r_i^{O2}$ directly. No new cell-block inversion is required. A different penalty matrix would generally require recomputing the local inverse.

The O2 target and all sensor values are rebuilt from the current cell-average conservative state at every residual evaluation. They remain fixed during the reconstruction subiterations of that residual evaluation. The solve is therefore linear in $\mathbf r_i$ even though the residual-to-residual mapping is nonlinear in the evolving cell means.

O2 target and original shock gate

The selected O2 reference starts from an explicit Green–Gauss gradient. A face-based characteristic WBAP limiter acts on that gradient, and a facial positivity contraction may pull the O2 polynomial toward the cell mean. This same finalized polynomial supplies both the sensor values and $\mathbf r_i^{O2}$. Barth-based LVR remains available as a separately labelled ablation; unqualified “LVR v2” in this post means the WBAP-based path.

For an internal face $f$, let $L$ and $R$ identify the cells on its two sides, and let the unit normal $\mathbf n_f$ point from $L$ to $R$. The O2 polynomials provide face pressure $p_{L,f},p_{R,f}$, velocity $\mathbf v_{L,f},\mathbf v_{R,f}$, and sound speed $c_{L,f},c_{R,f}$. Define

$$ j_{p,f}=\frac{2|p_{R,f}-p_{L,f}|} {\max(p_{L,f}+p_{R,f},\varepsilon_{p,f})}, $$$$ j_{c,f}=\frac{\max\left(0, (\mathbf v_{L,f}-\mathbf v_{R,f})\mathbin{\cdot}\mathbf n_f\right)} {\max((c_{L,f}+c_{R,f})/2,\varepsilon_{c,f})}. $$

Symbol	Definition
$f$	one internal face
$L,R$	cells on the left and right of $f$ relative to $\mathbf n_f$
$\mathbf n_f$	unit face normal directed from $L$ to $R$
$p_{L,f},p_{R,f}$	limited-O2 pressures reconstructed to face $f$
$\mathbf v_{L,f},\mathbf v_{R,f}$	limited-O2 velocities reconstructed to face $f$
$c_{L,f},c_{R,f}$	sound speeds derived from the limited-O2 face states
$j_{p,f}$	nondimensional pressure-jump indicator
$j_{c,f}$	nondimensional normal-compression indicator; it is zero in expansion
$\varepsilon_{p,f},\varepsilon_{c,f}$	small scale-aware denominator guards, not tuning parameters

For a nonnegative indicator $z$, its cubic start-to-full response is

$$ S(z;z_0,z_1)= \begin{cases} 0,&z\le z_0,\\ q^2(3-2q),&z_0\lt z\lt z_1,\\ 1,&z\ge z_1, \end{cases} \qquad q=\frac{z-z_0}{z_1-z_0}. $$

Here $z_0$ is the threshold below which the response is exactly zero, $z_1$ is the threshold at which the response reaches one, and $q$ is the normalized position between them. Let $\mathcal F(i)$ be the internal faces incident on cell $i$. The original product-gate response is

$$ a_i=\max_{f\in\mathcal F(i)}\left\{ \alpha_{max}S(j_{p,f};j_{p0},j_{p1})S(j_{c,f};j_{c0},j_{c1}) \right\}. $$

Symbol	Definition
$a_i$	original pressure-jump/compression product-gate value in cell $i$
$\mathcal F(i)$	set of internal faces incident on cell $i$
$j_{p0},j_{p1}$	pressure-jump start and full-response thresholds
$j_{c0},j_{c1}$	compression start and full-response thresholds
$\alpha_{max}$	configured upper bound for the penalty fraction

The product makes the gate exactly zero when either pressure jump or compression is absent. This suppresses activation at a pure contact, shear layer, or expansion.

LVR v2 near-shock correction

The original product gate left a weakly covered shoulder beside the Lax right-moving shock because the strongest pressure jump and strongest compression occurred on different faces. Threshold and $\alpha_{max}$ tuning could not repair this geometric offset. LVR v2 therefore detects a shock-supported pressure extremum next to the already-detected core.

Let $p_i$ be the cell-average pressure. A neighbour $j\in\mathcal N(i)$ is pressure-equal when

$$ |p_j-p_i|\le\tau_p\max(|p_i|,|p_j|,\varepsilon), $$

where $\tau_p$ is a relative tolerance and $\varepsilon=10^{-30}$ only guards zero pressure in the comparison. Pressure-equal neighbours are omitted from the extremum calculation. Let $\mathcal N_v(i)$ be the remaining pressure-varying neighbours. The normalized extremum magnitude is

$$ e_i= \begin{cases} \dfrac{p_i-\max_{j\in\mathcal N_v(i)}p_j} {\max\left(|p_i|,\left|\max_{j\in\mathcal N_v(i)}p_j\right|,\varepsilon\right)}, &p_j\lt p_i\ \text{for every }j\in\mathcal N_v(i),\\[1.0em] \dfrac{\min_{j\in\mathcal N_v(i)}p_j-p_i} {\max\left(|p_i|,\left|\min_{j\in\mathcal N_v(i)}p_j\right|,\varepsilon\right)}, &p_j\gt p_i\ \text{for every }j\in\mathcal N_v(i),\\[1.0em] 0,&\text{otherwise}. \end{cases} $$

The first branch detects a strict local maximum among pressure-varying neighbours; the second detects a strict local minimum. The correction is not seeded in boundary-adjacent cells. Let $n_{eq,i}$ be the number of pressure-equal face neighbours. The correction requires $n_{eq,i}\ge m_t$. The parameter $m_t$ is a neighbour count, not a number of halo layers.

The original gate in neighbouring cells supplies the shock-support value

$$ s_i=\max_{j\in\mathcal N(i)}\frac{a_j}{\alpha_{max}}. $$

The instantaneous extremum response, retained response, and correction seed are

$$ r_i=S(e_i;e_0,e_1)S(s_i;s_0,s_1), $$$$ r_i^{ret}=\eta\frac{\alpha_i^{old}}{\alpha_{max}}S(s_i;s_0,s_1), $$$$ c_i=\alpha_{max}\max(r_i,r_i^{ret}). $$

The final LVR v2 penalty fraction is

$$ \alpha_i=\max\left(a_i,c_i,\max_{j\in\mathcal N(i)}c_j\right). $$

Symbol	Definition
$p_i,p_j$	cell-average pressures in cells $i$ and $j$
$\tau_p$	relative tolerance used to classify pressure-equal neighbours
$\mathcal N_v(i)$	neighbours whose pressure is not pressure-equal to $p_i$
$e_i$	normalized strict pressure-extremum magnitude
$n_{eq,i}$	number of pressure-equal face neighbours
$m_t$	minimum pressure-equal neighbour count required for a correction seed
$s_i$	nearby support from the original product gate, normalized by $\alpha_{max}$
$e_0,e_1$	extremum start and full-response thresholds
$s_0,s_1$	shock-support start and full-response thresholds
$r_i$	instantaneous correction response
$\eta$	retained fraction from the preceding reconstruction/RHS update
$\alpha_i^{old}$	corrected penalty fraction from that preceding update
$r_i^{ret}$	response retained while current shock support remains present
$c_i$	correction seed before its one-neighbour extension
$\alpha_i$	final LVR v2 penalty fraction used in the reconstruction update

Because $c_i\gt0$ requires original-gate support in a face neighbour, a seed can be at most one face-graph hop from the original support ${i:a_i\gt0}$. The final neighbour maximum extends that seed by one more hop. The v2 correction can therefore reach at most two face-neighbour hops from the original detected region, but it is not a uniform two-layer dilation: the first-hop cell must pass every extremum, support, boundary, and equal-neighbour test. The selected runs set the separate generic gateHaloLayers value to zero.

The canonical v2 parameters are:

Mathematical symbol	Configuration field	Value	Meaning
$j_{p0},j_{p1}$	`pressureJumpStart`, `pressureJumpFull`	0.04, 0.16	pressure-jump start and full response
$j_{c0},j_{c1}$	`compressionStart`, `compressionFull`	0.03, 0.13	compression start and full response
$\alpha_{max}$	`alphaMax`	0.50	maximum penalty fraction
$e_0,e_1$	`oscillationExtremumStart`, `oscillationExtremumFull`	0.001, 0.008	extremum start and full response
$s_0,s_1$	`oscillationSupportStart`, `oscillationSupportFull`	0.02, 0.15	nearby shock-support start and full response
$\eta$	`oscillationRetention`	0.95	retained response fraction
$\tau_p$	`oscillationTransverseTolerance`	$10^{-8}$	pressure-equal relative tolerance
$m_t$	`oscillationMinTransverseNeighbours`	2	required pressure-equal neighbour count
$n_h$	`gateHaloLayers`	0	extra generic face-graph halo depth

The equal-neighbour guard deliberately recognizes the locally one-dimensional extruded Lax topology. It is experimentally validated but is not yet a mesh-independent multidimensional shock criterion.

Formulation progress

Stage	Result
Initial penalty study	Established the exact $\mathbf A_i$-weighted convex pull and showed that cached block inverses remain valid.
O2 target comparison	Replaced the initial Barth-only target with a selectable characteristic WBAP target; the WBAP target became canonical.
Derivative-weight tuning	Compared Factorial and HQM_OPT reconstruction functionals; HQM_OPT reduced the Lax shoulder oscillation and improved the cylinder residual behavior.
Product-gate tuning	Tuned $j_{p0},j_{p1},j_{c0},j_{c1},\alpha_{max}$ on Lax and the Mach-20 cylinder, while retaining exactly zero penalty in smooth cells.
LVR v2 correction	Added the shock-supported extremum correction, retention, equal-neighbour guard, and bounded seed extension.
Flux and case extension	Tested Rusanov, Roe M8/M9, Sod, Toro 123, two-shock, Shu–Osher, and the cylinder.
Multidimensional production	Completed aligned and unstructured Double Mach reflection, the 13-method DM240U matrix, and both structured and matched-cell-count Delaunay 2-D Riemann comparisons.

The detailed development reports are initial validation, parameter tuning, WBAP O2 retuning, HQM_OPT joint tuning, shock-band correction, and extended v2 validation. These links require access to the private study repository.

Lax shock-band reduction

At $t=0.15$ on the maintained 400-cell Lax tube, the original product gate left a pressure overshoot/undershoot on the constant state immediately beside the right-moving shock. The corrected v2 gate expanded from five to nine active cells and nearly flattened this shoulder without changing the curved Mach-20 cylinder result.

Method	Wrong-direction pressure excursion	Wrong-direction density excursion	Downstream left-plateau pressure range	Density $L_1$	Active cells
Fully limited P3 3WBAP	0	0	0.07173	not reduced in this study	all
Original product LVR, WBAP O2, HQM_OPT	0.11452	0.04356	0.09046	$4.208\times10^{-3}$	5
LVR v2, WBAP O2, HQM_OPT	0.00686	0.00279	0.00495	$4.284\times10^{-3}$	9
LVR v2, WBAP O2, Factorial	0.00808	0.00313	0.01502	$5.725\times10^{-3}$	9

For this right-moving shock, the nearly constant region immediately to its left is the downstream, post-shock plateau. For HQM_OPT, the correction reduces the pressure excursion by 94.0% and that plateau’s pressure range by 94.5%, while density $L_1$ rises only 1.8%. On the Mach-20 cylinder, corrected and original HQM_OPT histories are numerically identical: final absolute density $L_1$ residual $1.405\times10^{-8}$, 684 active cells, no wall or far-field activation, a two-cell centerline shock, and stagnation density 6.11134.

Extended compact-case validation

The extended study used exact cell-averaged Riemann solutions and an independent 20,000-cell characteristic WENO5 reference for Shu–Osher. Roe M8 and M9 reduce Lax, Sod, two-shock, and Shu–Osher errors relative to Rusanov, but both Roe variants stall on the Toro 123 double-rarefaction case with the tested positivity configuration. Rusanov remains the robust cylinder choice.

Representative density $L_1$ values for LVR v2/HQM_OPT are:

Case	Rusanov	Roe M8	Roe M9
Lax	0.004310	0.003594	0.003602
Sod	0.001491	0.001282	0.001285
Toro 123 double rarefaction	0.004974	stalled	stalled
Two-shock	0.09827	0.09648	0.09630
Shu–Osher	0.03055	0.02341	0.02359

The v2 correction is intentionally narrow: it fixes the Lax weak-gate gap, remains inactive on the curved cylinder bow shock, and does not turn the whole post-shock Shu–Osher wave train into an O2 solution.

Canonical 13-method matrix

The thtj1 campaign compares 13 configurations:

Explicit GG O2 with WBAP limiting;
LVR v2 with WBAP O2 target, HQM_OPT weights, $\alpha_{max}=0.5$;
the same with $\alpha_{max}=1$;
LVR v2 with WBAP O2 target, Factorial weights, $\alpha_{max}=0.5$;
the same with $\alpha_{max}=1$;
LVR v2 with Barth O2 target, HQM_OPT weights, $\alpha_{max}=0.5$;
the same with $\alpha_{max}=1$;
LVR v2 with Barth O2 target, Factorial weights, $\alpha_{max}=0.5$;
the same with $\alpha_{max}=1$;
fully limited P3 3WBAP with HQM_OPT weights;
fully limited P3 3WBAP with Factorial weights;
fully limited P3 CWBAP with HQM_OPT weights;
fully limited P3 CWBAP with Factorial weights.

The compact suite contains the Mach-20 cylinder plus Lax, Sod, Toro 123, two-shock, and Shu–Osher: 78 runs in total. Every run completed. The two tables below present every case twice. The first layout overlays all 13 methods in one plot. The second uses a controlled $2\times2$ layout to isolate method family, derivative weights, $\alpha_{max}$, and O2-target limiter.

All 13 methods in one layout

Each row gives the full case and a detailed view. Shock-tube details include states on both sides of the selected wave. Toro 123 has no shock, so its detail crosses the right rarefaction front. The Shu–Osher detail focuses on the post-shock entropy-wave train. For the steady cylinder, the complementary detail is the absolute density $L_1$ residual history because it distinguishes convergence floors that are hidden by the final centerline profile.

Case	Full view, all 13 methods	Detailed view, all 13 methods
Mach-20 cylinder
Lax shock tube
Sod shock tube
Toro 123 double rarefaction
Two-shock problem
Shu–Osher

The separate all-method Shu–Osher shock-front detail is also retained:

Controlled $2\times2$ factor layouts

Each $2\times2$ figure uses the same four comparisons:

Explicit O2, LVR v2, 3WBAP, and CWBAP with HQM_OPT weights;
HQM_OPT versus Factorial derivative weights;
LVR v2 with $\alpha_{max}=0.5$ versus $\alpha_{max}=1$;
WBAP-based versus Barth-based LVR v2 O2 targets.

Case	Full-view $2\times2$ layout	Detailed-view $2\times2$ layout
Mach-20 cylinder
Lax shock tube
Sod shock tube
Toro 123 double rarefaction
Two-shock problem
Shu–Osher

The separate $2\times2$ Shu–Osher shock-front detail is:

Reading the matrix

The principal observations are:

Explicit O2 is robust but visibly diffusive.
WBAP/HQM_OPT LVR v2 with $\alpha_{max}=0.5$ gives the flattest selected Lax downstream state while remaining sharper than Explicit O2.
Barth-based targets broaden strong shocks more visibly.
High-order LVR and conventional P3 limiters preserve substantially more Shu–Osher wave amplitude than Explicit O2.
The cylinder residual histories separate methods whose final centerline profiles look similar but whose convergence floors differ substantially.

Double Mach reflection

The DMR sequence tests shock geometry, the Mach-reflection triple point, the downstream slip line, wall jet, primary vortex, and Kelvin–Helmholtz roll-up. The triple point is the junction of the incident shock, reflected shock, and Mach stem. The slip line issues downstream from this junction.

Development campaign

Aligned UniformDM240 contains 230,400 quadrilateral cells. Its Rusanov and Roe M9 sweeps reached $t=0.25$. LVR produces a smooth triple point and delayed slip-line roll-up; halving the time step preserves that result. Fully limited 3WBAP produces more small-scale activity but also a more irregular triple point. The current interpretation is a perturbation-seeding hypothesis: 3WBAP’s irregular shock junction seeds stronger downstream disturbances, whereas LVR leaves the high-order slip line unlimited but supplies it with a quieter initial perturbation.

Replacing the aligned mesh with the nominal $h=1/240$ Delaunay DM240U mesh produces 609,468 triangles. Both LVR variants then resolve a long coherent roll-up train while their gate stays concentrated on the shock system. This shows that LVR does not inherently suppress smooth shear-layer instability. The test changes topology, orientation, perturbation spectrum, and cell count, so it does not isolate mesh noise or prove mesh-converged accuracy.

DM240U method	Wall time	Final density range	Validation
Explicit GG O2/WBAP	3,028 s	$[1.37913,22.49664]$	complete
LVR v2/HQM_OPT	11,713 s	$[1.33734,22.46597]$	reconstructed completion after supervisor loss
LVR v2/Factorial	15,874 s	$[1.33547,22.83554]$	complete
P3 3WBAP/HQM_OPT	26,148 s	$[1.31685,22.17635]$	complete
P3 3WBAP/Factorial	27,807 s	$[1.30258,22.00308]$	complete

Canonical 13-method DM240U production on `thtj1`

All 13 accepted rows completed on the same 609,468-cell Delaunay mesh with unrotated Roe M9, ESDIRK4, 625 steps of $4\times10^{-4}$ to $t=0.25$, 56 MPI ranks, cell-centered VTKHDF output, and no point output. Final local and remote snapshot hashes agree. The all-method figures use one shared density range $1\le\rho\le24$. The normalized LVR gate is $\chi=\alpha/\alpha_{max}$, so $\chi=0$ is an inactive penalty and $\chi=1$ is the configured maximum penalty.

Two-dimensional Riemann problem

Structured 400 x 400 production

The structured campaign uses 160,000 quadrilateral cells on $[0,1]^2$, unrotated Roe M9, ESDIRK4, 3,200 steps of $2.5\times10^{-4}$ to $t=0.8$, and 56 MPI ranks on thtj1. Both LVR v2 rows use the WBAP O2 target and $\alpha_{max}=0.5$; the references are fully limited P3 3WBAP. All four runs completed with exit code zero, 3,200 final-step markers, finite positive final states, empty validation-error lists, and nonempty cell-centered VTKHDF snapshots.

Delaunay 246U production

The unstructured control uses $h=1/246$, 160,420 counterclockwise triangles, 80,703 nodes, and the same domain, initial states, boundary name, flux, time step, final time, and four spatial methods. The cell count differs from the structured reference by only 420 cells.

All four rows completed 3,200 steps at $t=0.8$, returned exit code zero, retained finite positive density and energy, reported no validation errors, and wrote nonempty final cell-centered VTKHDF snapshots. Every snapshot and run record was copied back with matching remote and local SHA-256 hashes.

Method	Job	State at refresh	Slurm elapsed	Minimum $\rho$	Minimum $\rho E$
LVR v2/WBAP/HQM_OPT	`11790760`	complete	01:56:23	0.120155	0.229760
LVR v2/WBAP/Factorial	`11790784`	complete	01:54:34	0.119598	0.231224
P3 3WBAP/HQM_OPT	`11790785`	complete	12:05:34	0.125520	0.247447
P3 3WBAP/Factorial	`11790786`	complete	03:59:54	0.119668	0.236326

The fixed method order and shared range $0.1\le\rho\le1.8$ make the four final cell-centered density fields directly comparable.

The last completed row can be audited through these remote files on thtj1:

Slurm output: /fs2/home/advance/zhy/studies/limited_variational_reconstruction/2026-09-28_riemann2d_400_pilot/slurm_246u/riemann2d-246u-11790785.out
solver standard output: /fs2/home/advance/zhy/data/lvr/riemann2d_246u_m9_20260928/raw/3wbap_hqm_20260928T154338Z_thtj1-246u-m9-t08_11790785/stdout.log
absolute residual CSV: /fs2/home/advance/zhy/data/lvr/riemann2d_246u_m9_20260928/raw/3wbap_hqm_20260928T154338Z_thtj1-246u-m9-t08_11790785/riemann2d_.log
run record: /fs2/home/advance/zhy/studies/limited_variational_reconstruction/2026-09-28_riemann2d_400_pilot/records/3wbap_hqm_20260928T154338Z_thtj1-246u-m9-t08_11790785.json

Main findings

The $\mathbf A_i$-weighted penalty is algebraically compatible with the existing block Jacobi/GS/SOR machinery. It produces an exact convex target without rebuilding the cached cell inverse.
A characteristic WBAP-limited O2 target is materially better than the Barth target for the selected strong-shock comparisons, although the Barth path remains useful as an ablation.
The original pressure-jump/compression product gate localizes shocks and avoids pure expansions and shear layers, but its two indicators can be spatially offset beside a strong shock.
LVR v2’s shock-supported extremum correction closes that Lax gate gap and reduces the shoulder oscillation by about 94% without changing the curved cylinder bow-shock solution.
LVR v2 preserves convergence advantages over fully limited high-order schemes in the steady cylinder and is consistently much cheaper than fully limited 3WBAP in the current multidimensional runs.
The DMR results show that a quiet LVR triple point can delay slip-line roll-up, while the unstructured-mesh control shows that LVR still resolves substantial Kelvin–Helmholtz structure when perturbations are supplied by the mesh. Vortex count alone is not an accuracy metric.
The structured and completed unstructured 2-D Riemann contours show that the derivative-weight choice changes the developed interaction substantially, even when the LVR gate remains concentrated on discontinuities.

Limitations and next steps

Replace the equal-pressure-neighbour count with a geometric tangential smoothness test based on the limited-O2 pressure-gradient direction. This is needed for a mesh-independent multidimensional v2 gate.
Repeat Lax and multidimensional cases on refinement sequences to measure the physical width of the corrected band and verify that plateau flatness is not tied to the current spacing.
Repeat the structured and Delaunay 2-D Riemann comparisons on a refinement sequence to separate derivative-weight sensitivity from mesh-scale effects.
Separate DMR perturbation seeding from dissipation using matched degrees of freedom, time-resolved slip-line growth, and one common imposed disturbance.
Retain the Toro 123 positivity failure as a hard screen for Roe variants; gate widening must not be accepted by shock-tube appearance alone.
For Shu–Osher, assess post-shock phase, amplitude, $L_1/L_2$, total variation, and gate overlays together.

Complete figure atlas

The atlas below embeds every top-level PNG generated by the studies used in this post. It intentionally excludes duplicate PDFs and internal rendering tiles.

Initial method validation (10 figures)

Figure	Figure

Initial parameter tuning (2 figures)

Figure	Figure

Uniform-start cylinder convergence (1 figure)

Figure	Figure

WBAP O2 reference retuning (7 figures)

Figure	Figure

HQM_OPT joint tuning (5 figures)

Figure	Figure

LVR v2 shock-band gate (3 figures)

Figure	Figure

LVR v2 extended validation (13 figures)

Figure	Figure

Double Mach reflection development (41 figures)

Figure	Figure

Canonical 13-method matrix (30 figures)

Figure	Figure

Two-dimensional Riemann campaigns (4 figures)

Figure	Figure

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

Selector in the repaired source	Meaning	Added production frequency \(\Delta\)
`SAVersion=0`, default	Corrected SA-R	\(2\min(0,s-\omega)\)
`SAVersion=9001`	Bad historical version 0, for reproduction	\(2\min(0,\omega-s)\)
`SAVersion=1`	Oldest retained legacy formulation	\(\min(0,\sqrt{2}s-\omega)\)