---
title: Pressure-Equilibrium Preserving Method
url: https://www.emergentmind.com/topics/pressure-equilibrium-preserving-method
type: topic
---

# Pressure-Equilibrium Preserving Method

A pressure-equilibrium preserving method is a numerical discretization for the Euler or related compressible flow equations that guarantees discrete preservation of pressure equilibrium states—i.e., solutions in which velocity and pressure are constant or satisfy an equilibrium constraint, while other variables (such as density or species concentrations) may have strong spatial variation. These methods are critical in avoiding nonphysical “pressure oscillations” or “pressure blips” at contact interfaces, material boundaries, or flows with large density ratios, particularly in low-Mach or multi-material regimes. Pressure-equilibrium preservation (PEP) is established through careful design of fluxes, reconstruction operators, or source-term discretizations that enforce the invariance of pressure and velocity equilibria in the fully discrete update, typically down to machine precision.

## 1. Mathematical Criteria for Pressure-Equilibrium Preservation

The core requirement for PEP is that if the initial numerical solution satisfies a pressure (and possibly velocity) equilibrium condition, then the discretization should ensure that pressure and velocity remain invariant under the evolution equations, regardless of density, composition, or other advected scalars. For compressible Euler systems, a prototypical pressure equilibrium state satisfies
$$
u(x) \equiv U, \qquad p(x) \equiv P, \qquad \rho(x) \;\text{arbitrary}.
$$
For a scheme to be PEP, the update of the pressure (either as a conservative variable, a primary state variable, or via the equation of state) must yield $\partial_t p_n = 0$ at every node/cell if $p_n \equiv P$ and $u_n \equiv U$ at the previous time, for arbitrary local density [2407.03299].

Formally, the semi-discrete update should yield, for all cells or nodes $n$,
$$
\frac{d}{dt} p_n = 0, \quad \frac{d}{dt} u_n = 0 \quad \text{if initialized in pressure- and velocity-equilibrium.}
$$
The specifics of these constraints depend on whether the update uses the conservative variables $(\rho, \rho u, \rho E)$, the pressure evolution form, or an alternative variable projection.

## 2. Design Mechanisms in Modern Schemes

### 2.1 Flux Construction

Pressure-equilibrium preservation is tightly linked to the structure of the flux functions in finite volume (FV), finite difference (FD), or discontinuous Galerkin (DG) methods. Classical schemes may fail to preserve pressure equilibrium due to the separate interpolation of primitive or conservative variables, causing arithmetic-mean inconsistencies (e.g., in the energy flux) and violation of the algebraic linkage imposed by the equation of state. A rigorous algebraic criterion for PEP at the flux level states that, for ideal gases and second-order central or split forms, the energy flux should interpolate the density by harmonic mean rather than arithmetic mean to fulfill
$$
\widetilde \rho \cdot \widetilde{\rho^{-1}} = 1,
$$
for all face averages between adjacent cells. For instance, the energy flux may be written as
$$
F_{\rho E}^* = \rho_H \overline{e} \overline{u} + \cdots, \qquad \rho_H = \frac{2 \rho_L \rho_R}{\rho_L + \rho_R}, \quad \overline{e} = \frac{e_L + e_R}{2},
$$
where $L$, $R$ denote left/right states at the face [2407.03299]. This minimal modification ensures discrete preservation of pressure at all time steps.

### 2.2 Equilibrium-Projection and Pressure-Based Formulations

Alternative methodologies use equilibrium projection or pressure-based formulations. For DG schemes, one approach is to reconstruct or project the state vector onto equilibrium manifolds characterized by constant pressure (and possibly velocity), computing face fluxes in terms of these projected states [2211.16254, 2501.12532]. For example, one can define a pressure-projected state vector $\tilde{y}$ by:
$$
\tilde{y}(y, \tilde{p}) = (\rho v, \rho \tilde{u}(C, \tilde{p}) + \frac{1}{2} \rho v^2, C),
$$
where $\tilde{u}(C, \tilde{p})$ is chosen to ensure the pressure is equal to the polynomial projection $\tilde{p}$ in each cell [2211.16254].

A related strategy replaces the total energy equation with a pressure evolution equation, directly evolving $P$ with a DG method and supplementing with energy correction terms for semidiscrete conservation [2501.12532].

### 2.3 Source Term Discretization in Balance Laws

For systems with nontrivial source terms (e.g., Euler–gravity, shallow water with topography), equilibrium preservation hinges on discretizing source terms and flux gradients in a manner that exactly cancels at hydrostatic or other equilibrium solutions. This is often realized by:

- Rewriting the source terms using known equilibrium identities (e.g., replacing $-\rho \phi_x$ by $-(\rho/\rho^e)\partial_x p^e$).
- Using entropy-conservative or equilibrium-consistent flux differences for spatial derivatives.
- Carefully constructing nodal or cellwise source terms to match the discretized pressure gradients, ensuring that for any interpolant of the equilibrium state, the discrete residual vanishes [2503.00553, 2402.01131, 2507.13284].

## 3. Algorithmic Structures and Implementation

Several algorithmic structures for PEP can be identified:

| Methodology                        | Key Discrete Mechanism                              | Applicability                    |
|------------------------------------|-----------------------------------------------------|----------------------------------|
| Harmonic-mean-based FV/FD fluxes   | Facewise harmonic-mean for $\rho$ in energy flux    | Compressible Euler (ideal gas)   |
| Pressure-projection at faces       | Project local state to fixed $p$, reconstruct $u$   | Multicomponent, real fluids      |
| Pressure evolution DG              | Replace $E$-update by $P$-update, correct energy    | Multicomponent, arbitrary EOS    |
| Well-balanced nodal DG             | Entropy-conservative fluxes, equilibrium-based src  | Gravity/coupled source systems   |
| Equilibrium-variable projection    | Approximate DG space in equilibrium variables $V$   | Euler, shallow water, Ripa       |
| EC-WENO / EC-MRWENO reconstructions| Unified weights ensure $(v,p)$ const in interface   | Cut-cell, multi-material flow    |

In most cases, such mechanisms can be implemented as modular modifications of the flux evaluation or state-reconstruction routines in existing solvers, without the need for a change in mesh, time stepping, or limiter subroutines [2407.03299, 2507.19966].

## 4. Theoretical Properties and Well-Balancedness

The PEP property is closely related to the broader concept of well-balancedness—exact preservation of nontrivial equilibria (steady, possibly nonzero-velocity, or nonuniform pressure states) in balance laws.
- For the Euler–gravity system, well-balanced nodal DG achieves machine-precision preservation of hydrostatic equilibrium for any prescribed profile,
provided a careful equilibrium-based discretization of both fluxes and sources is applied [2503.00553].
- In DG schemes for general hyperbolic balance laws, constructing the polynomial approximation space in terms of equilibrium variables (e.g., $V=(K, m, \varepsilon)$, with $K=p/\rho^\gamma$) and using hydrostatic reconstruction at cell faces guarantees that the discrete scheme remains at equilibrium when so initialized [2402.01131].
- Pressure-equilibrium preservation is ensured in multi-material sharp-interface methods even under geometric uncertainties by advancing geometric moments and using equilibrium-compatible reconstructions [2507.19966].

The discrete well-balanced property has been proven for a variety of systems and methods, including arbitrary-order DG for both hydrostatic and moving equilibria [2402.01131, 2507.13284], and for cut-cell methods with moving interfaces [2507.19966].

## 5. Practical Implications, Limitations, and Extensions

PEP methods have significant implications for accuracy, robustness, and stability:
- Elimination of spurious pressure oscillations (“pressure blips”) at contact interfaces, enabling stable advection of strong density or composition gradients [2407.03299, 2512.04450].
- Machine-precision preservation of equilibrium across time steps in both low- and all-Mach regimes [2005.04296, 2503.00553].
- Accurate resolution of interface problems in multi-material or multi-phase flows, robust to geometric perturbations—an essential property for cut-cell and level-set-based interface tracking [2507.19966].
- High-order accuracy in smooth flow regions, with only mild (often second-order) degradation precisely at material interfaces [2507.19966, 2507.13284].

Limitations include:
- For real-gas, multi-component, or nonideal EOS, the construction of the dual mean in the flux or equilibrium-projection procedure must be carefully adapted [2512.04450]. The harmonic mean for density suffices only for ideal gases.
- For conservative versus pressure-based forms, the latter may require correction terms to recover strict global conservation, especially at the fully discrete level [2501.12532].
- Shock capturing and nonlinear stability are not automatically guaranteed by PEP alone; appropriate Riemann solvers or limiters are still necessary in the presence of strong discontinuities [2407.03299].

Extensions include pressure-robustness in incompressible Stokes/VEM contexts, where using divergence-preserving reconstruction operators in test and right-hand sides eliminates velocity error dependence on irrotational forces and achieves true pressure-independence [2007.00298, 2002.01830].

## 6. Impact and Representative Applications

Pressure-equilibrium preserving methods have been deployed in a wide spectrum of contexts, including:
- Compressible multi-fluid advection and interface tracking [2512.04450, 2507.19966];
- Low-Mach number atmospheric and astrophysical flows with complex source terms [2402.01131, 2503.00553];
- Chemically reactive or multi-component detonation problems, where pressure uniformity is crucial for correct wave speeds [2211.16254];
- Incompressible and mixed Stokes problems requiring robust velocity fields irrespective of pressure magnitude [2007.00298, 2002.01830];
- Shallow water and extended moment-equation models with complicated moving equilibria [2507.13284].

Numerical validation demonstrates that PEP methods prevent the build-up of artificial pressure oscillations and preserve equilibrium in challenging test cases across a variety of applications and mesh types, offering both machine-precision equilibrium and competitive—often optimal—convergence rates away from interfaces or discontinuous source features.

Source: https://www.emergentmind.com/topics/pressure-equilibrium-preserving-method