Reconstruction 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.

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