Wave-appropriate reconstruction of compressible flows: physics-constrained acoustic dissipation and rank-1 entropy wave correction

Abstract: The wave-appropriate reconstruction approach decomposes the reconstruction procedure into characteristic wave families, centralizing non-acoustic waves to minimize dissipation while retaining an upwind bias for acoustic waves. In previous implementations, the acoustic upwind parameter $η_a$ was fixed at its maximum value of $1.0$; however, this choice is conservative and motivated a systematic search for the minimum value that is robust across flow regimes. To this end, the CFD solver is treated as a black box within a bounded scalar minimization, which minimizes an accuracy objective for the subsonic inviscid TGV subject to a stability constraint enforced by the supersonic viscous TGV. Because the wave-appropriate framework leaves $η_a$ as the sole degree of freedom, the optimization converges in approximately 25 evaluations. The resulting optimal values generalize without retuning across a wide range from subsonic turbulence to hypersonic flows with shocks and contact discontinuities. The second contribution focuses on eliminating the need for an explicit contact-discontinuity detector, which is commonly required in flows involving both shock waves and contact discontinuities. In such cases, the reconstruction deficiency appears solely within the entropy characteristic wave and can be corrected by a rank-1 update along the entropy right eigenvector. The proposed algorithm relies only on the Ducros sensor and is limiter-agnostic, facilitating direct use in other schemes, such as WENO. This approach reduces wall time by $29$--$41\%$ compared to full characteristic decomposition. To further demonstrate the method's generality, introducing a controlled acoustic bias exclusively to the normal momentum in a KEP scheme eliminates spurious vortices in periodic shear layers, confirming that the acoustic stability mechanism operates independently of the discretization framework.

• The paper establishes a wave-appropriate reconstruction method that decomposes flow characteristics to tailor dissipation, retaining upwind bias only for acoustic waves.
• It optimizes the acoustic upwind bias (ηa) via bounded minimization to robustly capture turbulence while reducing excessive numerical dissipation.
• A rank-1 entropy wave correction is introduced to accurately resolve contact discontinuities without explicit sensors, cutting computational cost by up to 41%.

Wave-Appropriate Reconstruction of Compressible Flows: Physics-Constrained Acoustic Dissipation and Rank-1 Entropy Wave Correction

Introduction and Motivation

The paper presents a systematic development and empirical optimization of wave-appropriate reconstruction (WAR) schemes for the simulation of compressible flows within the finite-volume framework. The central concept is a characteristic-wave decomposition of the reconstruction step, enabling selective application of dissipation: upwind bias is retained only for acoustic waves, while shear and entropy/contact waves are handled with minimal or no dissipation. This physically informed selectivity addresses the well-established trade-off in high-order shock-capturing schemes—resolving small-scale turbulence without sacrificing stability near shocks and contacts. Prior works in the series established the separation of wave families and motivated centralization of non-acoustic waves; the current manuscript closes two open questions: (1) the minimum robust acoustic upwind dissipation, and (2) the optimal algebraic form of entropy/contact treatment without contact sensors.

Wave-Appropriate Reconstruction and Physics-Constrained Parameterization

The WAR methodology performs interface value reconstruction in characteristic space, leveraging the eigenstructure of the compressible Euler equations. The five characteristic fields correspond to: two acoustic modes (u-c, u+c), an entropy mode, and two shear/vortical modes. The key design principle is differentiated reconstruction strategies based on the wave type:

• Acoustic waves: reconstructed with an upwind blend parameterized by $\eta_a$, historically defaulted to 1.0.
• Shear/vortical waves: centrally reconstructed ($\eta=0.5$) except near shocks.
• Entropy wave: selectively limited; sharp interfaces employ the Tangent of Hyperbola for INterface Capturing (THINC).

Unlike previous frameworks that empirically tuned multiple parameters governing dissipation and sensor thresholds, this WAR design leaves $\eta_a$ as the sole free parameter in smooth regions. Its optimization is thus a bounded scalar constrained minimization, yielding rapid convergence with minimal computational budget.

Physics-Constrained Acoustic Dissipation and Empirical Optimization

Using the inviscid and viscous Taylor-Green Vortex (TGV) as calibration and stability benchmarks, the acoustic upwind bias $\eta_a$ is optimized through Brent's bounded minimization. The strategy is:

• Objective: Minimize the time-integrated deviation of grid-resolved turbulent kinetic energy from a reference high-order linear scheme.
• Constraint: Retain stability in both subsonic and supersonic, inviscid and viscous TGV regimes.

Key empirical outcomes:

• For the third-order WAR scheme (WA-3), $\mathbf{\eta_a^* = 0.54}$.
• For the fifth-order WAR scheme (WA-5 and derivatives), $\mathbf{\eta_a^* = 0.6010}$.

Both values are transferred unchanged to all subsequent test cases—turbulent, shocked, contact-rich—confirming their robustness and that the minimized dissipation is sufficient but not excessive. Notably, $\eta_a^*$ values are only incrementally above the symmetry-breaking central value (0.5) and far below the conventional upwind bias of 1.0.

The optimized nonlinear $N$th-order WAR schemes are shown to match, or outperform, reference linear $(N\!+\!2)$th-order upwind schemes on all smooth flow benchmarks.

Rank-1 Entropy Wave Correction and Computational Efficiency

A second major contribution is the obviation of explicit contact sensors in the treatment of entropy waves. By examining the eigenvector structure, it is demonstrated that the reconstruction error at contacts is solely a rank-1 perturbation in entropy space. Thus, a rank-1 update along the entropy right eigenvector suffices to correct conservative reconstruction errors near contacts, without the need for empirical or case-by-case detector tuning.

Algorithmic Details:

• In smooth regions (Ducros sensor inactive), conservative variable upwind/central blending is performed: $\eta_a^*$ for density, normal momentum, and energy; $\eta=0.5$0 for tangential momentum.
• The entropy characteristic is projected, run through an MP5 limiter (or any monotonic ENO/WENO), and a minimal rank-1 update is applied to the conservative state.
• In shocked regions (Ducros sensor exceeding threshold), classical full characteristic WAR is invoked.

This mechanism (dubbed WA-CR for "wave-appropriate with conservative reconstruction and rank-1 correction") leads to 29–41% wall-time reduction across a suite of high-dimensional test problems. The algebraic rank-1 update is limiter-agnostic, supporting both MP5 and WENO smoothness choices.

Extension to Kinetic-Energy-Preserving and Entropy-Stable Schemes

Most prior high-fidelity ILES strategies deploy central discretizations with dissipative upwinding activated near detected shocks. The manuscript extends WAR’s applicability by demonstrating that the acoustic upwind bias is the essential minimal stabilization component also in KEP/entropy-preserving schemes. Implementing the upwind bias only in the normal momentum flux (not the full characteristic system) suffices to suppress shear-layer instability (“spurious vortices”) that affect pure central schemes—even in the absence of shocks. This evidence underscores the independence of the acoustic stability mechanism from details of the underlying (central) discretization.

Numerical Results and Comparative Performance

A broad suite of canonical and challenging compressible flow benchmarks is addressed: inviscid and viscous TGV (subsonic/supersonic), periodic shear layers, Rayleigh-Taylor instability, explosion/shock-entropy interaction, double Mach reflection, Riemann problems, shock-bubble interactions, and viscous shock tubes.

Highlights:

• Optimized WAR with $\eta=0.5$1 resolves turbulent kinetic energy to the same scales as, or beyond, linear high-order upwind baselines.
• WAR schemes (WA-5, WA-CR, WA-WENO-CR) match the finest solution structure of classical WENO/TENO/ALDM competitors and often exhibit superior performance at lower computational cost.
• In flows with strong contacts (Rayleigh-Taylor, shock-bubble, slip-line dominated Riemann), the rank-1 entropy correction achieves sharp, oscillation-free density jumps without missing physical secondary instability.
• WA-CR’s wall-clock speedup is most pronounced in multi-phase/material interface problems and flows with spatially intermittent shocks/contacts, precisely due to WAR’s sensor-based path selection.
• KEP plus minimal acoustic bias matches WAR performance in the absence of background dissipation but fails without it, sharply delineating the necessity of the targeted upwinding.

Implications, Limitations, and Future Perspectives

The WAR framework, in its current instantiation, strictly relies on the conservative variable system and characteristic-space separation. Extension to primitive-variable reconstruction or other non-conservative variable sets would require additional eigenstructural analysis. The approach delivers generality across orders, limiters, and discretization frameworks, relying on physical interpretation over data-driven parameter fitting.

Theoretically, the value of $\eta=0.5$2 is identified by nonlinear stability boundaries empirically; deriving analytical or semi-analytical criteria relating WAR’s stability threshold to instabilities in the underlying PDE (e.g., Kelvin-Helmholtz or acoustic instabilities) is a compelling direction for research. There is also prospect for adaptivity: local or flow-dependent variation of $\eta=0.5$3 and further integration with machine-learned or adjoint-based optimization of dissipative mechanisms.

Conclusion

The paper rigorously quantifies the minimal acoustic upwind bias necessary for robust, accurate compressible flow simulation and devises an algebraically optimal, computationally efficient, and empirically validated algorithm for characteristic wave-appropriate reconstruction. The rank-1 entropy wave correction is shown to yield sensor-free, robust contact capture with significant efficiency improvements. The reported methodology achieves the global minimum-dissipation configuration subject to stability, is transferable without retuning, and generalizes across ILES, KEP, and classical shock-capturing contexts. This forms a concrete advance toward systematically physics-informed, scalable, and generalizable high-order compressible flow solvers.