eMOOD: Entropy-Based Multidimensional Optimal Order Detection for High-Order Shock-Capturing Finite-Volume Schemes

1. Introduction

• High-order linear schemes produce spurious oscillations near discontinuities, motivating methods that control oscillations without abandoning high-order accuracy.The introduction cites Godunov’s theorem [4, 5] and Harten on monotonicity and TVD limitations.Source: §1 · p. 1
• MOOD advances with a high-order linear scheme, detects oscillations a posteriori, and recomputes only troubled cells using progressively more robust lower-order schemes.The paper notes that typically fewer than 5% of cells require recomputation.Source: §1 · p. 2
• Finite-volume MOOD implementations mostly retain variants of the relaxed DMP strategy, while alternative troubled-cell indicators have appeared in non-FV implementations.The Euler-equation extension relaxed the original DMP criterion heuristically to avoid flagging smooth extrema.Source: §1 · p. 2
• eMOOD replaces the classical DMP-based MOOD criterion with a compact numerical-entropy-production indicator targeting shocks, contact discontinuities, and under-resolved structures.It improves troubled-region localisation and avoids special treatment of smooth extrema.Source: §1 · pp. 2–3
• The method is implemented in UCNS3D and assessed on unstructured two-dimensional meshes using the classical MOOD cascade and a MOOD-CWENO hybrid.The hybrid retains inexpensive linear reconstruction in smooth regions and activates nonlinear CWENO reconstruction only where needed.Source: §1 · p. 3

2.1. Explicit Finite Volume Method

• Cell-average updates are obtained by integrating the conservation law over each cell and summing numerical fluxes across its faces.Face fluxes are evaluated at Gaussian quadrature points using HLL or HLLC Riemann solvers.Source: §2.1 · p. 3

2.1.1. High-order in time

• Unless otherwise specified, test problems use CFL = 0.75 and a third-order SSP Runge–Kutta method for time integration.The time step is selected according to the CFL condition using cell length and an estimate of maximum wave speed.Source: §2.1.1 · p. 4

2.1.2. High-order in space

• A least-squares polynomial fit on a stencil of at least K direct and indirect neighbours provides the spatial reconstruction, but a single stencil yields a linear scheme that can oscillate.By Godunov’s theorem [4], such a scheme cannot be TVD.Source: §2.1.2 · p. 4

2.2. CWENO reconstruction

• The final reconstruction is a weighted combination of polynomials fitted to the candidate stencils, with each weight depending on its polynomial’s smoothness indicator.A small constant ϵ = 10^-14 is added to avoid division by zero.Source: §2.2 · pp. 4–5
• CWENO replaces WENO’s large directional stencils with smaller stencils and first-order reconstruction polynomials, while modifying the central polynomial.This addresses the computational expense of multiple polynomial fits and smoothness-indicator evaluations.Source: §2.2 · p. 5
• On unstructured meshes, the study uses stencils of 2K cells and follows the stencil-selection algorithm of Tsoutsanis et al..The paper reports CWENO as meaningfully faster while offering accuracy and robustness comparable to or better than the original WENO method.Source: §2.2 · p. 5

2.3. MOOD

• MOOD accepts a high-order update when a cell passes the checks; otherwise, it repeats the update with MUSCL and then a first-order scheme if needed.The repeated fluxes and time-step updates involve the cell and its neighbours.Source: §2.3 · p. 5
• Parallel MOOD requires additional communication and synchronisation for average updates contaminated by fluxes from inadmissible cells, so fewer and more clustered flagged cells are favourable.Scattered flagged cells produce an unpredictable communication pattern and more neighbouring cells to handle.Source: §2.3 · p. 5

2.4. MOOD-(C)WENO Hybrid

• The hybrid applies full CWENO reconstruction in MOOD’s second step to avoid relying on lower-order methods that can reduce accuracy or miss higher-order phenomena.The first-step linear reconstruction is reused as CWENO’s central stencil; only smaller directional-stencil reconstructions and weights are then needed.Source: §2.4 · p. 5
• Figure 2 presents the symbolic flow chart of the hybrid MOOD-CWENO method for comparison with Figure 1.Source: Figure 2 · p. 6
• Linear reconstruction remains sufficient in most cells, while the hybrid is designed to retain traditional CWENO’s robustness and order of accuracy.This is expected to make it more efficient than applying CWENO throughout.Source: §2.4 · p. 6

2.5. Switch Criteria

• The switch criterion seeks to minimize flagged cells while maintaining accuracy, and favors quick computation using cell-local or limited-neighbour data.Source: §2.5 · p. 6

2.5.1. PAD

• The reference PAD and NAD checks reject inadmissible states and use a relaxed discrete maximum principle to detect new extrema.PAD flags NaN components or negative density or pressure; NAD checks the relaxed DMP condition.Source: §2.5.1 · p. 6

2.5.2. Entropy Production Indicator

• The proposed indicator flags a cell when its entropy change exceeds β, rescaling that change to account for mesh resolution and time-step dependence.Entropy change is the absolute difference between the potential solution’s entropy and that of the previous time step.Source: §2.5.2 · p. 6
• The threshold uses a smaller β1 on the first pass and a larger β2 on the second, selected for each test case using characteristic velocity, density, and pressure.For two-gas tests, the entropy is a volume-fraction-weighted mixture of the two ideal entropies.Source: §2.5.2 · p. 7

2.5.3. Entropy Production Scaling

• The scaling investigation tests whether the trigger works across large cell-size ranges and whether thresholds can transfer from coarse to refined meshes.The deformed mesh experiment targets the first requirement; resolution-independent thresholds are described as desirable.Source: §2.5.3 · p. 7, Figure 3 · p. 7
• The scaling study tested five entropy-production strategies on the Shu-Osher problem, tuning β1 and β2 independently for each strategy.The mesh-refinement study used four Forward Facing Step meshes and compared four strategies after matching coarse-mesh results.Source: §2.5.3 · pp. 7–8
• On a heavily deformed mesh, no cell-size scaling was the best of five tested strategies; multiplying by cell area missed the smallest cells, while dividing by area flagged them prematurely.Using characteristic length instead of area reduced, but did not eliminate, those flaws.Source: §2.5.3 · p. 8, Figure 4 · p. 8
• The recommended Δs·h scaling reduces the flagged-cell fraction as the Forward Facing Step mesh is refined, unlike Δs/h, which increases it.Δs·A reduces flagging too quickly and fails to flag shock regions; Δs·h is used in subsequent mesh-refinement tests.Source: §2.5.3 · p. 9, Figure 5 · p. 9

3.1. Shu-Osher Test

• The MOOD-CWENO hybrid with the entropy trigger is closest to the reference CWENO5 solution in the Shu-Osher shocked-wave cross-section.All five methods produce very similar overall solutions.Source: Figure 6 · p. 10, §3.1 · p. 10

3.2. Double Mach Reflection

• The entropy-trigger MOOD-CWENO hybrid closely matches CWENO5 in the Double Mach Reflection test, while entropy-trigger MOOD has a slightly more diffused front shock.The hybrid with the classical trigger produces an incorrect front-shock position and shape at the wall.Source: Figure 7 · p. 11, §3.2 · p. 12

3.3. Forward Facing Step

• All five methods produce practically indistinguishable Forward Facing Step profiles, closely matching a reference computed on a mesh with four times higher resolution.Source: Figure 9 · p. 13, §3.3 · p. 13

3.4. Helium Bubble - Shock Wave Interaction

• The four MOOD-based methods yield effectively indistinguishable helium-bubble results, though a higher-density ring appears around the bubble relative to CWENO5.The finer mesh visualizes increased accuracy with quadrupled resolution.Source: Figure 10 · p. 14, §3.4 · p. 14

3.5. Triple Material Point

• The four MOOD-based methods give similar Triple Material Point results, while the hybrid yields a less-diffused material interface left of the vertex than pure MOOD.Relative to the reference, increased density appears around the material interface, especially beneath the reflected wave on the right.Source: Figure 12 · p. 16, §3.5 · p. 16

4.1. Shock detection

• The entropy trigger flags cells along shocks and discontinuities, unlike the scattered classical-trigger pattern, and flags cells inside shocks rather than smooth regions.The same pattern occurs in multi-phase simulations, where the linear blend of phase entropies flags cells correctly in both phases without spurious flags.Source: §4.1 · pp. 16–18

4.2. MOOD-CWENO hybrid accuracy

• The hybrid can improve accuracy over traditional MOOD, including sharper material interfaces, but the difference is limited when fewer than 5% of cells are flagged.With an artificially reduced entropy threshold, the hybrid's accuracy improvement may help when trigger configuration is imperfect or prohibitively difficult.Source: §4.2 · pp. 18–19

4.3.1. Timing

• Runtime comparisons include initialization, the simulation loop, and output, with code, mesh, and parallelization choices held as constant as possible across methods.Tests ran on a 16-core workstation and a 128-core Archer2 partition, with four runs averaged for each test combination.Source: §4.3.1 · p. 19

4.3.2. Triggers comparison

• MOOD with the entropy-production trigger is up to 10% faster than the equivalent simulation with the classic trigger.Source: §4.3.2 · p. 20

4.3.3. MOOD-CWENO hybrid

• Up to 20% speed-up over classic CWENO5 is reported for the MOOD-CWENO hybrid.The authors note that communication algorithms optimized for the hybrid could potentially widen the difference.Source: §4.3.3 · p. 21
• Pure MOOD is faster than the hybrid because flagged cells use less involved lower-order schemes, while the hybrid can compensate through accuracy and robustness in many scenarios.Source: §4.3.3 · p. 21

• eMOOD uses numerical entropy production as a compact, physically motivated alternative to the classical DMP-based troubled-cell criterion, improving localisation while avoiding excessive smooth-region activation.Source: §5 · p. 21
• The entropy-based detector reduces flagged cells by up to 75% across the considered benchmarks while preserving original MOOD robustness and non-oscillatory properties.Its compact, local formulation also improves memory-access locality and reduces communication overhead.Source: §5 · p. 21
• The MOOD-CWENO hybrid achieves speed-ups of up to 25% while maintaining comparable accuracy and robustness.It retains high-order reconstruction in flagged regions, improving fine-scale flow resolution and reducing sensitivity to detector imperfections.Source: §5 · pp. 21–22
• The methods were validated on diverse compressible-flow benchmarks, and future work will extend them to direct and indirect ALE formulations on high-order unstructured meshes in two and three dimensions.Source: §5 · p. 22