Reaction-region indicator: a progress update

Thu, 10 Sep 2026 16:00:00 +0800

When chemistry, diffusion, and shocks occupy very different parts of a flow, a single integration treatment can be an awkward compromise. This work explores a cell-local reaction-region indicator that chooses how much of each cell is treated with tightly coupled chemistry and flow, and how much is handled by Strang splitting.

The goal is deliberately practical: use coupling where the interaction is strongest, while retaining the benefits of splitting where it is appropriate. It is a numerical-method selector, not a claim to identify a universal physical reaction zone.

The idea

The selector combines three dimensionless signals:

local chemical activity;
local diffusion activity; and
a shock gate based on the pressure jump.

Chemistry and diffusion must both be active before the coupled fraction becomes large. The shock gate suppresses coupling at a strong detonation shock, where splitting can be preferable for propagating the discontinuity. The result is a continuous field rather than a binary mask.

Mathematical definition

Let $\Delta t_{\mathrm{phys}}=\Delta t_{\mathrm{code}}L_0/U_0$ be the physical duration of a solver step. For cell $i$, the chemical activity is

$$ a_i = \Delta t_{\mathrm{phys}}r_{\mathrm{chem},i}, \qquad r_{\mathrm{chem},i} = \left[ \sum_{k=1}^{N_s} \left(\frac{\dot\omega_{k,i}W_k}{\rho_i}\right)^2 +\left(\frac{|\dot q_i|}{\rho_i c_{v,i}T_{\mathrm{scale},i}}\right)^2 \right]^{1/2}. $$

Here $\dot\omega_k$ is the net molar production rate, $W_k$ the molecular weight, $\rho$ the density, $c_v$ the constant-volume specific heat, and

$$ \dot q_i=-\sum_{k=1}^{N_s}\dot\omega_{k,i}\bar h_{k,i}, \qquad T_{\mathrm{scale},i}=\max(T_i,T_{\mathrm{floor}}). $$

Thus $r_{\mathrm{chem}}$ is a non-negative inverse chemical time scale: it combines species conversion with the fractional temperature-source rate, so neither can cancel the other.

The diffusion activity is

$$ b_i=\Delta t_{\mathrm{phys}}\frac{D_{\max,i}}{L_{\mathrm{grad},i}^{2}}, $$

where $D_{\max,i}$ is the largest mixture species diffusivity and the local gradient length is reconstructed from face neighbours:

$$ L_{\mathrm{grad},i}^{-1} =\max_{j\in\mathcal N(i)} \left[ \left( \frac{T_j-T_i}{d_{ij}\max(T_i,T_j,T_{\mathrm{floor}})} \right)^2 +\sum_{k\in\mathcal A_{ij}} \left( \frac{Y_{k,j}-Y_{k,i}} {d_{ij}\max(Y_{k,i},Y_{k,j},10^{-3})} \right)^2 \right]^{1/2}, $$$$ L_{\mathrm{grad},i} =\max\!\left(\frac{1}{L_{\mathrm{grad},i}^{-1}},\Delta x_i\right). $$

$\mathcal N(i)$ is the face-neighbour set, $d_{ij}$ is the distance between cell centres, and $\mathcal A_{ij}$ retains species whose larger adjacent mass fraction is at least $10^{-3}$. The lower bound $\Delta x_i$ prevents a subcell gradient length.

The pressure-jump sensor and its shock gate are

$$ h_i=\max_{j\in\mathcal N(i)} \frac{|p_j-p_i|}{\max(|p_i|,|p_j|,p_\epsilon)}, \qquad g_h(h_i)=\frac{1}{1+(h_i/h_0)^4}. $$

The three inputs $a_i$, $b_i$, and $h_i$ are dimensionless. In particular, $g_h\approx1$ in a smooth flame and tends to zero across a sufficiently strong pressure jump.

For positive thresholds $a_0$, $b_0$, and exponent $p$, each activity is saturated by

$$ \operatorname{sat}(z;z_0,p) =\frac{\bigl(\max(z,0)/z_0\bigr)^p} {1+\bigl(\max(z,0)/z_0\bigr)^p}, $$

and the coupled score is

$$ C_i= \operatorname{sat}(a_i;a_0,p) \operatorname{sat}(b_i;b_0,p) g_h(h_i). $$

The original saturation is recovered by $(a_0,b_0,p)=(1,1,1)$, for which $\operatorname{sat}(z)=z/(1+z)$.

From score to split treatment

For the figures below, the displayed coupled fraction is the complement of the Strang fraction:

$$ c = 1 - \chi, \qquad \chi = 0\ \text{for fully coupled integration}, \qquad \chi = 1\ \text{for Strang splitting}. $$

The original logistic map is

$$ c_i^*=\sigma\!\left( \frac{C_i-C_0-w\ln b_s}{w} \right), \qquad \sigma(z)=\frac{1}{1+e^{-z}}, $$

where $C_0$ is the score midpoint, $w$ the transition width, and $b_s$ a Strang-preference factor. The historical settings used $C_0=0.005$, $w=0.001$, $h_0=0.08$, and $b_s=1$.

The compact-tail Hill alternative is

$$ c_i^*=\frac{C_i^n}{C_i^n+(C_0b_s)^n}, $$

with exponent $n>0$. Unlike the logistic map, it gives exactly $c_i^*=0$ when $C_i=0$. Optional face-neighbour expansion then applies

$$ c_i^{(m+1)}= \operatorname{clip}_{[0,1]} \left[ \max\!\left( c_i^{(m)}, \eta\,g_h(h_i)\max_{j\in\mathcal N(i)}c_j^{(m)} \right) \right], $$

where $\eta\in[0,1]$ is the retention per pass. The tuned comparison used $n=3$, two passes, and $\eta=0.65$.

Finally, with $\chi_i^*=1-c_i^*$, endpoint tolerances enforce exact limiting treatments:

$$ \chi_i= \begin{cases} 0, & \chi_i^*\le\epsilon_c,\\ 1, & 1-\chi_i^*\le\epsilon_s,\\ \chi_i^*, & \text{otherwise}. \end{cases} $$

The thresholds and powers change the finite-step transition, not the physical state. Since $a_i$ and $b_i$ both scale with $\Delta t_{\mathrm{phys}}$, $\chi$ is intentionally a time-step-dependent numerical-method selector.

Mixed integrator

Write the semidiscrete reactive-flow system as $\dot u=F(u)+S(u)$, with $F$ the non-reactive flow operator and $S$ the chemical source. The mixed method uses the same frozen $\chi_i$ field in every part of one physical step:

$$ u^{n+1}= B_{\chi S}\!\left(\frac{\Delta t}{2}\right) A_{F+(1-\chi)S}(\Delta t) B_{\chi S}\!\left(\frac{\Delta t}{2}\right)u^n. $$

In words: each cell receives its $\chi_i$-weighted chemistry in two source-only half steps, while the full implicit flow step carries the complementary $(1-\chi_i)S_i$ source contribution. The selector is evaluated from the physical-step entry state and remains fixed through both half steps and all ODE stages.

What the profiles show

The one-dimensional H2/O2 flame example below gives an intuitive view. In flames, the coupled region follows the thin part of the thermal and chemical structure. In detonations it stays much narrower and does not simply cover the leading shock. Increasing the chosen time step broadens the selected region, which is expected: the indicator is responding to the numerical challenge of taking a larger step through a reactive, diffusive layer.

Figure — Frozen H2/O2 flame at three numerical time-step factors. The dotted black curve is normalized temperature; blue, orange, and green curves are the original selector’s coupled fraction $1-\chi$ at $\Delta t/\Delta t_0=0.01$, 1, and 1000. The physical profile is unchanged between curves. The very large factor is an intentionally formal sensitivity probe, included to show the selector’s limiting tendency rather than a usable solver step.

This is a useful distinction. The underlying chemical and diffusion rates are physical diagnostics, whereas the final selector is allowed to depend on the step size because it decides between numerical treatments. These plots keep the flow state fixed, so they reveal the selector’s behavior but do not by themselves prove trajectory accuracy.

One-dimensional a-priori profiles

The fixed-profile view is also useful at the level of the individual indicator ingredients. The flame and detonation panels below evaluate the same formula before a mixed trajectory is evolved. They answer a deliberately narrow question: where would the selector request coupled treatment for this stored thermochemical structure?

Figure — Fixed H2/O2 flame profile at $\Delta t_{\mathrm{code}}=2\times10^{-3}$. From top to bottom: temperature and pressure; major species; radical mass fractions; the chemical and diffusive one-step activities $a$ and $b$; their saturated factors, shock gate, and coupled score; and the final Strang fraction $\chi$ with its coupled complement $1-\chi$. The narrow coupled band is aligned with the flame transition. This is an a-priori classification of a fixed profile, not a propagation result.

Figure — Fixed H2/O2 detonation profile at $\Delta t_{\mathrm{code}}=4\times10^{-6}$. The upper panels show the shock-adjacent thermochemical transition and radicals; the next panels show $a$, $b$, saturation, the pressure-jump gate, and the resulting score. Although chemical activity rises near the leading structure, the shock gate collapses at the discontinuity, leaving the displayed split fraction overwhelmingly Strang-dominant. This is a frozen-state diagnostic, not an accuracy assessment.

One-dimensional a-posteriori profiles

The next two figures are different evidence: they are taken from completed mixed simulations. Each compares the nontrivial logistic and tuned selector fields on their own evolved states, so it shows the selector as it actually participated in the calculation. It must not be read as an independent test of the fixed-state calibration.

Figure — A-posteriori ESDIRK2 flame profiles at $\Delta t=4\times10^{-3}$, relative to the propagated front. The logistic result occupies the top row and the tuned result the bottom row. Left panels give temperature and pressure, centre panels give the resulting chemical and diffusive activities, and right panels give the shock sensor, coupled score, and $\chi$. Both mixed solutions retain a compact coupled interval around the flame transition, despite the different final states. This is a posterior record from a completed coarse-step calculation; it demonstrates behavior of the mixed treatments, not convergence.

Figure — A-posteriori U2R1 detonation profiles at $\Delta t=8\times10^{-6}$, relative to the propagated shock. Logistic and tuned treatments again occupy the upper and lower rows. Activity concentrates upstream of the discontinuity, while the shock sensor and score are sharply localized at the front; $\chi$ remains close to one over almost the entire plotted length. The plot makes the intended detonation behavior visible: mixed integration is localized rather than a blanket replacement of Strang splitting. It does not establish mesh or inner-solver convergence.

A two-dimensional detonation picture

The same ingredients were examined on a cellular detonation snapshot. The temperature field has a wrinkled reaction front; the coupled-fraction panel follows that front rather than filling the entire hot product region. The shock gate makes the leading discontinuity visibly different from the reactive structure behind it.

Figure — A deliberately compact two-panel view of the O4-Strang cellular-detonation snapshot: temperature at left and the aggressive-setting coupled fraction $1-\chi$ at right. Both panels use the same retained $x>0$ field; the isoline traces the corrugated front. The coupled request remains a narrow band behind that front rather than filling the hot products. Because this is an aggressive setting, it is a spatial illustration rather than a default recommendation.

The particular setting shown here is intentionally aggressive, so it is best read as a spatial demonstration rather than a recommended default. It also exposes the remaining tuning question: a useful selector must retain a sharp response to shocks while giving a sufficiently smooth transition around reaction layers.

The comparison below puts four restrained threshold-and-power choices on the same cellular-detonation snapshot. The legacy, moderate, and conservative settings remain almost entirely split at this step; the balanced candidate activates along the corrugated reaction front. It is a helpful visual reminder that tuning controls where coupled integration is requested, not just the peak value of a scalar score.

Figure — Coupled fraction $1-\chi$ for the legacy, moderate-transition, conservative, and balanced candidates at the same physical step and on the same $x>0$ cellular-detonation field. Only the balanced candidate visibly lights up a narrow corrugated front. The shared color scale makes the contrast a selector choice, not a change in the underlying flow state.

Time step matters in two dimensions as well. Holding the snapshot fixed, the legacy selector requests essentially no coupling at a very small step, remains near Strang splitting at the recorded base step, and becomes much more expansive at an intentionally extreme step. This sensitivity is expected for a method selector, but it also makes clear why threshold choices must be assessed together with the temporal discretization.

Figure — One fixed O4-Strang temperature field at left, followed by the legacy coupled fraction $1-\chi$ at $\Delta t/\Delta t_0=10^{-2}$, $1$, and $10^3$. The near-uniform dark middle panels are the expected small-step logistic tail, not missing data. At the deliberately extreme factor, coupled treatment expands through the cellular burned-gas structure, illustrating numerical-method sensitivity rather than an evolved trajectory.

Two-dimensional field galleries

The originally dense 12-panel maps have been redrawn as paired, readable pages. Each pair uses the retained $x>0$ domain of one archived H2/O2/Ar cellular-detonation snapshot: the first page shows composition and the two activity inputs; the second shows the pressure-jump gate, score, and two coupled-fraction outputs. These are frozen-state classifications, not new two-dimensional propagation calculations.

Baseline selector fields at the archived step

Figure — O2-coupled snapshot at $\Delta t_{\mathrm{code}}=2\times10^{-5}$. Top row: hydrogen and oxygen mass fractions; bottom row: $\log_{10}a$ and $\log_{10}b$, the chemical and diffusive one-step activities. The aligned corrugations locate the reactive cellular layer, but activity alone does not determine the requested coupled treatment.

Figure — Selector page for the same O2-coupled snapshot. Top row: pressure-jump gate $g_h$ and $\log_{10}C$; bottom row: logistic and tuned coupled fractions $1-\chi$, each on a 0–1 scale. The gate suppresses the leading discontinuity and the coupled-fraction maps remain close to zero at this archived step.

Figure — O4-coupled snapshot at the same code step. Hydrogen and oxygen composition are shown above the logarithmic chemical and diffusive activities. The view uses the same field convention as the O2 case, making the cellular structures comparable without turning different archived orders into accuracy rankings.

Figure — Selector page for the O4-coupled snapshot: shock gate and score above, logistic and tuned coupled fractions below. The wrinkled pressure discontinuity is visibly gated, while both output maps preserve only a thin, low-amplitude reaction-layer trace at this step.

Figure — O4-Strang snapshot at the same archived step. The composition maps and the two activity maps identify a cellular reaction zone behind the front. Shared layout and scales make this a visual counterpart to the O4-coupled snapshot, not a propagated-method comparison.

Figure — Selector page for the O4-Strang snapshot. The shock gate tracks the corrugated discontinuity and the score is concentrated behind it, yet the logistic and tuned coupled fractions are visually near zero because the score is below the switch scale at this $\Delta t$. This separates physical activity from the time-step-dependent numerical request for coupled integration.

Fixed-snapshot time-step response

In the next three montages, the left panel is the same unchanged temperature field and the other three panels are the legacy coupled fraction $1-\chi$ at $\Delta t/\Delta t_0=10^{-2}$, $1$, and $10^3$. The flows do not evolve between panels: only terms that explicitly contain the proposed time step are rescaled.

Figure — O2-coupled time-step scan. The first two coupled-fraction panels are nearly dark because the original logistic map stays near its small-step tail. At $10^3\Delta t_0$, bright contours spread through the cellular burned-gas structure. This broadening is the selector’s response to a much larger numerical update, not a statement that the whole region is a physical reaction zone. Open the full-resolution figure.

Figure — O4-coupled time-step scan. Temperature at left is fixed across all columns. The $10^{-2}$ and base-step fields again have only the small logistic tail; the $10^3$ panel selects a large part of the resolved cellular structure. Its stronger response than the O2 case follows from the stored local activities, not from a re-run of the detonation. Open the full-resolution figure.

Figure — O4-Strang time-step scan. The left temperature panel is unchanged, and the three right panels use the same legacy selector at successively larger step factors. The visually quiet base-step panel is quantitatively nonzero but remains far below a strongly coupled threshold; the extreme-step panel demonstrates why a calibration cannot be treated as independent of the temporal discretization. Open the full-resolution figure.

Figure — O4-Strang tuning comparison. Rows are the moderate-transition and maximum-feasible threshold-and-power candidates; columns pair the unchanged temperature field with coupled-fraction maps at $10^{-2}$, $1$, and $10^3$ times the base step. The moderate candidate stays nearly inactive at its calibration step, whereas the maximum-feasible candidate forms a narrow cellular-front band there. Both become much broader at the extreme factor, showing that threshold-and-power tuning shifts activation but does not remove the large-step tendency. Open the full-resolution figure.

Early end-to-end results

The first H2/O2 flame comparisons are encouraging on robustness. At coarser time steps, mixed treatments completed cases where a pure Strang calculation with one temporal method lost physical admissibility. In the completed comparisons, the mixed curves generally tracked the fully coupled result more closely than pure Strang.

Flame speeds

The independently computed reference is $s_{L,\mathrm{ref}}=2.2531\ \mathrm{m\,s^{-1}}$. Each entry below is the fitted speed in $\mathrm{m\,s^{-1}}$, followed by its signed relative error in parentheses. failed is a numerical failure, not an omitted measurement.

$\Delta t$ / treatment	ESDIRK2	U2R2	U2R1
$7.5\times10^{-4}$ / coupled	2.699 (+19.773%)	2.693 (+19.519%)	2.719 (+20.690%)
$7.5\times10^{-4}$ / Strang	2.755 (+22.265%)	2.477 (+9.937%)	2.739 (+21.553%)
$7.5\times10^{-4}$ / logistic	2.689 (+19.355%)	2.712 (+20.373%)	2.711 (+20.335%)
$7.5\times10^{-4}$ / tuned	2.696 (+19.650%)	2.682 (+19.040%)	2.715 (+20.501%)
$2\times10^{-3}$ / coupled	2.677 (+18.806%)	2.672 (+18.574%)	2.788 (+23.743%)
$2\times10^{-3}$ / Strang	3.103 (+37.738%)	failed	2.867 (+27.250%)
$2\times10^{-3}$ / logistic	2.643 (+17.284%)	2.628 (+16.624%)	2.715 (+20.515%)
$2\times10^{-3}$ / tuned	2.658 (+17.976%)	2.635 (+16.961%)	2.722 (+20.829%)
$4\times10^{-3}$ / coupled	2.627 (+16.597%)	2.610 (+15.852%)	2.853 (+26.633%)
$4\times10^{-3}$ / Strang	failed	failed	3.018 (+33.932%)
$4\times10^{-3}$ / logistic	2.629 (+16.706%)	2.590 (+14.953%)	2.782 (+23.478%)
$4\times10^{-3}$ / tuned	2.615 (+16.078%)	2.666 (+18.345%)	2.756 (+22.341%)

Detonation speeds

For detonation, the independent Chapman–Jouguet reference is $D_{\mathrm{CJ}}=2836.4\ \mathrm{m\,s^{-1}}$. Each entry is the fitted shock speed in $\mathrm{m\,s^{-1}}$, followed by its signed relative error.

$\Delta t$ / treatment	ESDIRK2	U2R2	U2R1
$2\times10^{-6}$ / coupled	2791.699 (-1.576%)	2759.400 (-2.715%)	2773.981 (-2.201%)
$2\times10^{-6}$ / Strang	2772.506 (-2.253%)	2753.164 (-2.935%)	2770.994 (-2.306%)
$2\times10^{-6}$ / logistic	2772.768 (-2.243%)	2753.164 (-2.935%)	2770.994 (-2.306%)
$2\times10^{-6}$ / tuned	2772.506 (-2.253%)	2753.164 (-2.935%)	2771.050 (-2.304%)
$4\times10^{-6}$ / coupled	2903.887 (+2.379%)	2784.194 (-1.841%)	2793.641 (-1.508%)
$4\times10^{-6}$ / Strang	2845.226 (+0.311%)	2842.500 (+0.215%)	2787.498 (-1.724%)
$4\times10^{-6}$ / logistic	2844.348 (+0.280%)	2842.500 (+0.215%)	2786.752 (-1.750%)
$4\times10^{-6}$ / tuned	2845.170 (+0.309%)	2842.500 (+0.215%)	2787.088 (-1.739%)
$8\times10^{-6}$ / coupled	3187.864 (+12.391%)	3721.374 (+31.201%)	2828.152 (-0.291%)
$8\times10^{-6}$ / Strang	3212.499 (+13.260%)	3785.872 (+33.475%)	2842.297 (+0.208%)
$8\times10^{-6}$ / logistic	3211.484 (+13.224%)	3789.052 (+33.587%)	2850.351 (+0.492%)
$8\times10^{-6}$ / tuned	3212.025 (+13.243%)	3787.225 (+33.522%)	2851.163 (+0.520%)

There is an important qualification: these are progress results, not a final accuracy claim. The flame calculations are not yet mesh-converged against the independent reference, and nonlinear solves often reached their iteration cap. The evidence is therefore strongest for the selector’s localized behavior and its potential to improve robustness; quantitative accuracy still needs better-converged, carefully controlled follow-up calculations.

Final choices comparison

The latest tuning screen reduces the earlier threshold, power, time-step, and spatial-pass variants to four named choices. Every curve was recomputed with the current threshold-and-power activity saturation, pressure-jump gate, Hill outer map, endpoint snapping, and optional face-neighbor expansion. The inputs remain frozen profiles and snapshots, so this evidence describes where the selector would request coupling—not propagated-solution accuracy.

Role	Chemical threshold	Diffusive threshold	Power	Spatial passes	Status
Default	1	1	1	0	Production default, unchanged
Broad transition	5.6234	1.7783	0.5	0	Primary next posterior candidate
Broad + 2 passes	5.6234	1.7783	0.5	2	Separate spatial-edge experiment
Upper-bound screen	0.001	0.17783	1	0	A-priori stress test only

The broad transition uses gentler saturation shoulders without adopting the large coupled plateau of the upper-bound screen. Keeping the two-pass case separate makes its mesh-neighbor dependency visible rather than folding it into a threshold recommendation.

Figure — Frozen H2/O2 flame selector profiles. The three data panels hold the same physical flame fixed at $\Delta t/\Delta t_0=0.1$, 1, and 10; the fourth panel is the shared legend. Gray shading is normalized temperature and colored curves are coupled fraction $1-\chi$. At the base step, the broad transition increases the active-zone mean from 13.62% to 20.07%; the upper-bound screen is intentionally much broader and is not a production recommendation.

Figure — Frozen H2/O2 detonation profiles at the same three time-step factors. The compact panels make the shock-adjacent pocket readable: at the base step, default selection is 0.00% in the active zone, while the broad transition selects 2.81% and the two-pass variant 3.65%. The profiles are a-priori selector footprints, not propagated detonation results.

Figure — Frozen GRI30 methane/air flame profiles. At the base step, the broad transition raises the active-zone mean coupled fraction from 15.28% to 28.20%; two passes change it only to 28.45%, consistent with their primarily edge-local action in a smooth one-dimensional flame. The upper-bound screen reaches 89.34%, which is why it remains a stress test only.

Figure — Frozen GRI30 methane/oxygen ZND profiles. The base-step broad transition raises the active-zone mean from 26.42% to 56.64%, while the upper-bound screen reaches 78.94%. The figure shows the chosen numerical treatment across a fixed structure; it does not validate a ZND trajectory.

Figure — Length-weighted active-zone mean coupled fraction versus time-step factor for the same four frozen profiles. Endpoint snapping makes sufficiently small steps exactly Strang rather than leaving a small logistic residue. The broad transition moves activation to smaller factors than the default but remains materially less aggressive than the upper-bound screen; the two-pass curve differs chiefly at coupled-region edges. At large steps, saturation makes the pressure-jump gate the dominant remaining limiter.

Figure — Broad-transition choice on the frozen O4-Strang cellular-detonation field, shown on the retained $x>0$ domain. Temperature occupies the upper-left panel; the other panels show coupled fraction at $\Delta t/\Delta t_0=0.1$, 1, and 10. The first two choices are exact Strang on this snapshot. At ten times the base step, 1.154% of the domain exceeds coupled fraction 0.1 along the wrinkled reaction front. This is a fixed-snapshot selector footprint, not a two-dimensional propagation result.

Figure — Upper-bound screen on that same fixed O4-Strang field and time-step sequence. Unlike the moderate broad transition, it already selects 1.098% of the domain above coupled fraction 0.1 at the base step and reaches 15.574% at ten times the step. This deliberately large footprint is useful as a robustness envelope, but is not a production setting.

Where this is heading

The primary next posterior experiment is now the broad transition with zero spatial passes. It is designed to smooth the edge of the selected region without converting most of a flame to fully coupled integration. The paired two-pass choice remains a separate spatial-continuity experiment. The main questions are straightforward: do these choices preserve shock-aware localization, improve stability at useful time steps, and avoid hiding errors that belong to the temporal method or mesh instead?

\(\Delta t\) / treatment	ESDIRK2	U2R2	U2R1
\(7.5\times10^{-4}\) / coupled	2.699 (+19.773%)	2.693 (+19.519%)	2.719 (+20.690%)
\(7.5\times10^{-4}\) / Strang	2.755 (+22.265%)	2.477 (+9.937%)	2.739 (+21.553%)
\(7.5\times10^{-4}\) / logistic	2.689 (+19.355%)	2.712 (+20.373%)	2.711 (+20.335%)
\(7.5\times10^{-4}\) / tuned	2.696 (+19.650%)	2.682 (+19.040%)	2.715 (+20.501%)
\(2\times10^{-3}\) / coupled	2.677 (+18.806%)	2.672 (+18.574%)	2.788 (+23.743%)
\(2\times10^{-3}\) / Strang	3.103 (+37.738%)	failed	2.867 (+27.250%)
\(2\times10^{-3}\) / logistic	2.643 (+17.284%)	2.628 (+16.624%)	2.715 (+20.515%)
\(2\times10^{-3}\) / tuned	2.658 (+17.976%)	2.635 (+16.961%)	2.722 (+20.829%)
\(4\times10^{-3}\) / coupled	2.627 (+16.597%)	2.610 (+15.852%)	2.853 (+26.633%)
\(4\times10^{-3}\) / Strang	failed	failed	3.018 (+33.932%)
\(4\times10^{-3}\) / logistic	2.629 (+16.706%)	2.590 (+14.953%)	2.782 (+23.478%)
\(4\times10^{-3}\) / tuned	2.615 (+16.078%)	2.666 (+18.345%)	2.756 (+22.341%)

\(\Delta t\) / treatment	ESDIRK2	U2R2	U2R1
\(2\times10^{-6}\) / coupled	2791.699 (-1.576%)	2759.400 (-2.715%)	2773.981 (-2.201%)
\(2\times10^{-6}\) / Strang	2772.506 (-2.253%)	2753.164 (-2.935%)	2770.994 (-2.306%)
\(2\times10^{-6}\) / logistic	2772.768 (-2.243%)	2753.164 (-2.935%)	2770.994 (-2.306%)
\(2\times10^{-6}\) / tuned	2772.506 (-2.253%)	2753.164 (-2.935%)	2771.050 (-2.304%)
\(4\times10^{-6}\) / coupled	2903.887 (+2.379%)	2784.194 (-1.841%)	2793.641 (-1.508%)
\(4\times10^{-6}\) / Strang	2845.226 (+0.311%)	2842.500 (+0.215%)	2787.498 (-1.724%)
\(4\times10^{-6}\) / logistic	2844.348 (+0.280%)	2842.500 (+0.215%)	2786.752 (-1.750%)
\(4\times10^{-6}\) / tuned	2845.170 (+0.309%)	2842.500 (+0.215%)	2787.088 (-1.739%)
\(8\times10^{-6}\) / coupled	3187.864 (+12.391%)	3721.374 (+31.201%)	2828.152 (-0.291%)
\(8\times10^{-6}\) / Strang	3212.499 (+13.260%)	3785.872 (+33.475%)	2842.297 (+0.208%)
\(8\times10^{-6}\) / logistic	3211.484 (+13.224%)	3789.052 (+33.587%)	2850.351 (+0.492%)
\(8\times10^{-6}\) / tuned	3212.025 (+13.243%)	3787.225 (+33.522%)	2851.163 (+0.520%)

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