Approximate Pressure-Equilibrium Preserving Method

• Approximate pressure-equilibrium preserving methods are numerical techniques designed to maintain steady pressure states and reduce spurious oscillations in compressible, multicomponent flows.
• They modify traditional flux formulations using corrections, equilibrium-variable reformulations, or harmonic averaging to ensure high-order accuracy and conservation.
• These methods are applied in multi-material simulations, turbulent mixing, and interface tracking, especially when dealing with nontrivial equations of state.

An approximate pressure-equilibrium preserving method (PEP) refers to a numerical discretization technique for hyperbolic or parabolic PDEs—often arising in compressible fluid dynamics, multicomponent flows, or balance-law systems—which is constructed to (i) discretely preserve stationary or slowly-varying pressure (mechanical) equilibria to machine precision (or with a controlled, minimal error) and (ii) systematically eliminate, or at least reduce, the spurious pressure oscillations that commonly arise when conservative numerical methods are applied at smooth or discontinuous material interfaces, particularly when nontrivial equations of state (EOS) and/or multiple chemical species are present. The methodology spans (but is not limited to) finite volume, finite element, discontinuous Galerkin (DG), virtual element, cut-cell, and enriched Galerkin contexts; it encompasses both strictly exact and high-order approximately preserving approaches. Key design criteria include maintaining conservation, high-order accuracy, and pressure/velocity, or more generally, equilibrium-variable preservation—often via special reconstruction, flux averaging, or algebraic correction strategies.

1. Fundamental Concepts and Motivation

Pressure-equilibrium preserving (PEP) methods originate from the observation that classical fully conservative discretizations of the Euler (or Navier–Stokes) equations, when applied to multicomponent or real-fluid flows, tend to generate severe spurious pressure oscillations near contact discontinuities, smooth interfaces, or in regions of variable EOS, even if the initial velocity and pressure are uniform. This deficiency is fundamentally due to incompatibility between the nonlinear thermodynamic relations and the discrete averaging (arithmetic, harmonic, or other) used for flux construction. In physical terms, genuine stationary solutions—mechanical equilibria with $p = p_0$ and $v = v_0$—should give rise to zero pressure evolution both analytically and numerically; failure to enforce this property at the discrete level leads to artificial pressure errors that corrupt long-time simulations and interface dynamics. PEP methods address this by ensuring that the discrete update respects the equilibrium structure of the continuum system, either exactly or up to a controlled, high-order truncation error, often with minimal modification of an existing conservative scheme (DeGrendele et al., 4 Dec 2025, Ching et al., 21 Jan 2025, Michele et al., 3 Jul 2024, Cai et al., 26 Jul 2025).

2. Discretization Frameworks Exhibiting Approximate Pressure-Equilibrium Preservation

Several families of methods have been developed to realize the pressure-equilibrium preserving property:

• Flux Correction/Modification (Finite Volume and Central Schemes): Methods such as APEC (Approximately Pressure–Equilibrium-Preserving Conservative) modify only the internal energy (or total energy) half-point at cell interfaces via explicit algebraic corrections dependent on EOS thermodynamic derivatives, ensuring that the discrete pressure-compatibility condition is nearly satisfied. High-order variants extend these corrections to multi-point stencils (DeGrendele et al., 4 Dec 2025).
• Equilibrium Variable Reformulation (Discontinuous Galerkin, Balance Laws): The "equilibrium-preserving space" approach rewrites the system in terms of variables for which the targeted equilibrium corresponds to a constant field. By discretizing in the new coordinates, the equilibrium is exactly preserved in the DG polynomial space, and fluxes are modified by reconstruction using "hydrostatic" variables, ensuring arbitrary equilibria (hydrostatic, isobaric, or moving) can be preserved (Zhang et al., 2 Feb 2024).
• Flux Mean Adjustment (Kinetic Energy/Enthalpy-Preserving Fluxes): A minimal modification involves replacing the arithmetic mean for selected variables (typically density in internal energy flux) by the harmonic mean, with dual mean constructs ensuring that for uniform $p$, $u$ the discrete fluxes do not generate pressure evolution. Such modifications are compatible with central-difference/finite volume schemes and their high-order extensions (Michele et al., 3 Jul 2024).
• Pressure-Based or Pressure-Evolution Formulations (DG): Certain DG schemes replace the total energy equation with a pressure-evolution equation on the discrete level. To retain conservation, energy-correction terms, constructed to enforce elementwise discrete energy balance, are added. Further modifications ensure that such correction terms do not destroy the pressure or velocity-equilibrium properties (Ching et al., 21 Jan 2025).
• Cut-Cell and Interface Tracking Schemes: In compressible multi-material flow, cut-cell approaches evolve both conserved variables and geometric moments of cut-cells synchronously, using equilibrium-compatible (EC) reconstruction (e.g., EC-variant MRWENO) and special redistribution steps to maintain pressure equilibrium even under geometric perturbations and mesh motion (Cai et al., 26 Jul 2025).

The following table organizes principal approximate pressure-equilibrium preserving frameworks:

Approach (Editor’s term)	Core Mechanism	Representative Reference(s)
APEC / Flux Correction (FV)	EOS-based $\rho e$ correction	(DeGrendele et al., 4 Dec 2025)
Equilibrium-variable DG/FE	DG in equilibrium variables V	(Zhang et al., 2 Feb 2024, Liu et al., 1 Mar 2025)
Harmonic/dual-mean central flux	Adjust mean in energy flux (FV/CD)	(Michele et al., 3 Jul 2024, DeGrendele et al., 4 Dec 2025)
Pressure-evolution DG with energy correction	Nonconservative pressure update + fix	(Ching et al., 21 Jan 2025)
Moment-consistent cut-cell with EC recon	Moment evolution + EC polynomial rec	(Cai et al., 26 Jul 2025)

3. Core Algorithmic and Analytical Features

The specific construction of PEP methods depends on context, but several principles recur:

• Discrete Equilibrium Consistency: The flux discretization is devised so that, for any cells $m$, $m+1$ with $$(p_m, u_m, Y_{k,m}) = (p_{m+1}, u_{m+1}, Y_{k,m+1})$$, the update yields $\partial_t p = 0$ (exact or truncated at high order).
• Thermodynamic Correction: In multicomponent or real-fluid flows, corrections use thermodynamic derivatives evaluated at constant pressure (e.g., $$\epsilon_k = \partial (\rho e)/\partial (\rho Y_k)|_{p}$$) in the formulation of fluxes for the internal energy or species variables.
• High-Order Consistency: For higher-order methods, correction is applied at every sub-interface in the stencil, preserving the formal order of accuracy while reducing leading-order spurious pressure errors. The correction coefficients systematically diminish the magnitude of pressure error terms in Taylor expansion (DeGrendele et al., 4 Dec 2025).
• Conservation: All schemes remain strictly conservative (mass, momentum, energy), or conservation is restored by explicit correction terms (as in energy-corrected pressure-DG).
• Extensibility and Efficiency: Minimal algorithmic overhead—typically only requiring changes in local flux averaging or periodic correction terms—enables efficient integration into preexisting codes.
• Provable Preservation Properties: Many schemes admit proofs (exact or approximate) that stationary equilibria (hydrostatic, isentropic, or prescribed by EOS) are maintained to machine accuracy, provided the spatial discretization, flux formulation, and time integration (typically via SSP-RK) are consistent with the correction (Cai et al., 26 Jul 2025, Zhang et al., 2 Feb 2024).

4. Representative Methodologies: Detailed Examples

APEC (Approximately Pressure–Equilibrium-Preserving Conservative): For the one-dimensional multicomponent Euler equations, the flux for internal energy at the interface is modified as

$$\hat{(\rho e)}_{m+1/2}^{\mathrm{APEC}} = \frac{\rho e_m + \rho e_{m+1}}{2} + \frac{1}{2} \sum_{k=1}^{N} \Bigl[ \epsilon_k|_m - \epsilon_k|_{m+1} \Bigr] \frac{ \rho Y_{k,m} - \rho Y_{k,m+1} }{2 }$$

where the $\epsilon_k$ are evaluated for each species and all quantities at cell centers. The species half-point flux is taken as simple arithmetic means. This construction yields an $O(\Delta x^2)$ error in equilibrium, but with a coefficient at least halved (and more rapidly decaying at higher order) compared to the standard conservative average, resulting in several orders of magnitude smaller spurious pressure oscillations (DeGrendele et al., 4 Dec 2025).

Equilibrium-Variable DG Approach: Transforming variables to $V$ such that $V$ is constant at equilibrium (e.g., $V = [p/\rho^\gamma,\, \rho u,\, \epsilon + \phi]$ in Euler–gravity) allows the DG semi-discrete system to preserve any target equilibrium exactly for arbitrary mesh and polynomial degree, independent of the non-polynomial character of the conservative variables (Zhang et al., 2 Feb 2024).

Harmonic-Mean PEP Fluxes: For second-order central fluxes (Kuya–Coppola–Pirozzoli), pressure-equilibrium is realized by constructing internal energy fluxes via

$F_{\rho e} = \mathrm{#_{harm}}(\rho) \cdot \mathrm{#_{arith}}(u) \cdot \mathrm{#_{arith}}(e)$

with the harmonic mean ensuring that the required discrete dual-mean constraint is satisfied for constancy in $u$ and $p$ (Michele et al., 3 Jul 2024).

Cut-Cell with Equilibrium-Compatible Reconstruction: In multi-material compressible flow, the merging of tiny cut-cells is performed with auxiliary transport of higher-order geometric moments, ensuring polynomials are transported exactly on a moving mesh. Reconstruction at interface is performed with EC polynomials: if all stencils within a merged cell belong to the set $G(v^*,p^*)$ (the constant pressure/velocity manifold), then reconstructed polynomials (including WENO variants) also belong to this set, ensuring the equilibrium is discretely preserved even under geometric perturbation (Cai et al., 26 Jul 2025).

5. Analysis of Errors, Validation, and Performance

Systematic reduction of spurious pressure errors is evidenced by both Taylor expansion and empirical results. For APEC, the relative $L_2$ pressure error $\varepsilon_p$ for a central finite-volume scheme on a moving interface in a uniform pressure/velocity field reduces from $O(10^{-1})$ (standard) to $O(10^{-6})$ (second-order APEC) and $O(10^{-12})$ (eighth-order APEC), achieving near-machine-precision pressure flatness in long-time advection (DeGrendele et al., 4 Dec 2025).

In equilibrium-variable DG schemes, errors in static or moving equilibria remain at roundoff for all mesh sizes and polynomials, and arbitrary equilibria (hydrostatic, isobaric, moving) are preserved exactly on general, possibly non-uniform, meshes (Zhang et al., 2 Feb 2024). For cut-cell methods, pressure errors remain $O(10^{-14})$ at interfaces with minimal conservation loss, while purely conservative or geometric conservation law–only variants show degradation in order and stability under geometric perturbations (Cai et al., 26 Jul 2025).

6. Applications and Limitations

PEP methods have been applied successfully to:

• Compressible multi-material and multi-component flows with arbitrary EOS or strong interface jumps
• Turbulent mixing and interface tracking in multiphase simulations
• Hydrostatic or isobaric equilibrium preservation under gravity (Euler–gravity systems)
• Advection and long-time evolution of multicomponent bubbles (1D, 2D, and 3D)
• Flows requiring robust preservation of pressure flatness during interface motion, including sharp- and diffuse-interface cut-cell flows

Limitations include the need for explicit knowledge of thermodynamic derivatives for the EOS (in APEC and related schemes), moderate additional computational cost for moment evolution or polynomial reconstruction (in cut-cell and high-order DG schemes), and possible difficulties integrating nonconservative pressure-based updates with limiters or shock-capturing procedures in the fully nonlinear regime. Extensions to non-polynomial or extremely stiff EOS may require tailored algebraic treatment (DeGrendele et al., 4 Dec 2025, Ching et al., 21 Jan 2025).

7. Connections, Extensions, and Research Directions

The approximate pressure-equilibrium preserving paradigm connects closely to:

• Pressure-robustness in (incompressible) finite element methods—where divergence and moment-preserving reconstruction eliminate spurious velocity errors due to pressure gradients (Kreuzer et al., 2020, Frerichs et al., 2020)
• Well-balanced schemes for balance laws, especially in the context of gravity, shallow water, and the Ripa model (Zhang et al., 2 Feb 2024, Liu et al., 1 Mar 2025)
• Entropy-conservative and positivity-preserving frameworks in DG and FV methods (Liu et al., 1 Mar 2025)
• Recent developments in exact preservation of interface and geometric robustness for front-tracking/cut-cell approaches (Cai et al., 26 Jul 2025)

Active research includes development of fully exact and high-order PEP schemes for arbitrary nonlinear EOS, integration with adaptive mesh refinement, and systematic extension to multidimensional, non-Cartesian, and dynamically moving mesh contexts. Further analytical work is exploring the connection between algebraic compatibility conditions for pressure preservation and entropy conservation, as well as the impact on stability and long-time accuracy for practical high-speed, multiphase, and turbulence-resolving flow regimes.