Interpolation Pressure Methods

• Interpolation pressure is the process of reconstructing pressure fields or jumps between discrete data points using methods like kernel, momentum-weighted, and spline interpolations.
• It improves regional error control in sound field reproduction and stability in CFD by reducing sampling burdens and enhancing numerical convergence.
• Applications span sound synthesis, incompressible flow, equation-of-state modeling, and planetary sciences, with techniques ensuring physical consistency and improved simulation accuracy.

Interpolation pressure denotes a class of constructions in which pressure, pressure-derived quantities, or pressure jumps are reconstructed between discrete observations, computational locations, thermodynamic states, or physical phases. The surveyed literature suggests that the term does not identify a single canonical method; rather, it covers kernel-based regionalization of acoustic pressure error, pressure-weighted or momentum-weighted interpolation on collocated CFD discretizations, interpolation of equations of state in the $P$–$\mu$ or $(T,\rho)$ planes, direct interpolation of measured pressure fields, and reconstruction of pressure jumps at immersed interfaces (Koyama et al., 2022, Kühl et al., 2022, Blaschke et al., 2018, Baturin et al., 2019, Sase et al., 2024, Facci et al., 7 Mar 2026).

Domain	Interpolated object	Representative construction
Sound field reproduction	Regional pressure error from control-point pressures	Kernel-induced weighting matrix in weighted pressure matching
Collocated and meshfree CFD	Face pressure/velocity coupling or virtual-point pressure	Pressure-weighted, momentum-weighted, or VIP interpolation
Equation-of-state modeling	Pressure between phases or mesh points	Trigonometric, causal, or Hermite-spline interpolation
Measurement-driven rendering and planetary science	Measured or site-adjusted pressures	Bilinear interpolation or hydrostatic/GCM correction

1. Acoustic pressure interpolation as regional error control

In sound field reproduction, interpolation pressure appears most explicitly in weighted pressure matching (WPM). Conventional pressure matching minimizes synthesis error only at a discrete set of control points inside a target region $\Omega$,

$$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$

so the region between control points is not taken into consideration. WPM replaces this pointwise objective by the regional integral

$$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$

and then uses kernel interpolation in an RKHS to reduce the regional criterion to a weighted least-squares problem at the control points,

$$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$

For the uniform kernel,

$\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$

the weighting matrix is induced by integration of the kernel interpolant over the target region; a directional kernel is also supported when prior direction information is available (Koyama et al., 2022).

The experimental configuration reported for WPM used 12 loudspeakers in a square $(2\,\mathrm m\times 2\,\mathrm m)$, 16 control points in an inner $(1\,\mathrm m\times 1\,\mathrm m)$ square, and a single plane wave as the desired field. The evaluation metric was SDR integrated over the region. Above $\mu$0, WPM and directional WPM outperformed conventional pressure matching; at $\mu$1, the reported SDRs were $\mu$2 for PM, $\mu$3 for WPM, and $\mu$4 for directional WPM. Error maps showed reduced errors between control points, not only at them (Koyama et al., 2022).

A further theoretical step links WPM to weighted mode matching (WMM). In that formulation, the weighting matrix for WPM is defined on the basis of kernel interpolation of the sound field from pressures at a discrete set of control points, while the weighting matrix in WMM is defined by regional integration of spherical wavefunctions. By infinite-dimensional harmonic analysis for estimating expansion coefficients from pressure observations, WPM is shown to be a special case of WMM. Experimentally, to obtain approximately $\mu$5 SDR at $\mu$6, WPM required 64 control points whereas PM required 196, which makes explicit that regional interpolation can change the sampling burden, not merely the optimizer (Koyama et al., 2023).

2. Pressure interpolation on collocated and meshfree flow discretizations

In incompressible-flow numerics, interpolation pressure is often inseparable from pressure–velocity coupling. The Virtual Interpolation Point (VIP) method stores pressure and velocity dofs on a single node set but enforces the discrete equations on virtual points arranged as a virtual staggered structure. Pressure at a virtual interpolation point $\mu$7 is reconstructed as

$\mu$8

and pressure gradients are evaluated through virtual east–west and north–south points. The method proves a discrete inf-sup condition,

$\mu$9

thereby establishing existence and stability of discrete Stokes solutions without an explicit staggered grid. In the viscous-flow formulation, the same virtual staggered structure and virtual local stencil were reported to give second-order spatial and temporal convergence in Taylor decaying vortices and benchmark agreement for lid-driven cavity, triangular cavity, circular-cylinder, and bumpy-cylinder flows (Park et al., 2014, Park et al., 2014).

For pressure-based FV solvers with collocated variable arrangements, momentum-weighted interpolation (MWI) generalizes Rhie–Chow-type coupling. The adjoint counterpart is especially delicate because adjoint momentum equations contain additional source terms. The discrete adjoint face-velocity rule is written as

$(T,\rho)$0

with source-term interpolation

$(T,\rho)$1

where the interpolation lever for adjoint momentum sources must be reversed relative to the primal. Correct adjoint MWI improved sensitivity accuracy against FD references and yielded about $(T,\rho)$2 fewer outer iterations in the tested case, whereas using the wrong lever degraded both convergence and accuracy (Kühl et al., 2022).

A distinct pressure-weighted interpolation was proposed for unsteady incompressible finite differences on collocated grids in general curvilinear coordinates. Standard Rhie–Chow interpolation uses current-pressure gradients at cell interfaces and introduces a first-order temporal error in unsteady flow. The revised interface velocity instead uses the pressure difference

$(T,\rho)$3

which yields second-order accuracy in time for the interface interpolation and for continuity satisfaction. The same study omitted cross-derivative terms from the pressure-correction Poisson equation and used simultaneous relaxation of velocity and pressure; the Poisson solve was reported to be up to about $(T,\rho)$4 faster in 3D problems, while kinetic-energy conservation remained excellent even on non-orthogonal grids (Yanaoka, 2023).

For Euler–Euler gas–solid flow, generalized momentum interpolation and face-to-center flux reconstruction extend the same logic to multiphase coupling. The face value is decomposed as

$(T,\rho)$5

and the complete pseudo-momentum reconstruction includes gas pressure, granular pressure, drag, time-discretization, and under-relaxation corrections. A complete first-order flux reconstruction then updates cell-centered velocities from the corrected face fluxes. The method was described as independent of the time step, the transient-term discretization scheme, the under-relaxation factor, and the shape of grid, and it suppressed high-frequency oscillations more effectively than the standard OpenFOAM solver, although the paper also attributes the remaining oscillations near abrupt variable changes to the insufficiency of current Taylor-expansion-based reconstructions (Liu et al., 2024).

3. Global pressure formulations, post-processed trajectories, and pressure jumps

A different use of interpolation pressure appears in three-phase compressible flow in porous media. The global-pressure formulation introduces

$(T,\rho)$6

so that the total volumetric flow obeys

$(T,\rho)$7

This formulation is fully equivalent to the original equations only for Total Differential (TD) three-phase data. Because experimental data are usually available only on the boundaries of the ternary diagram, the paper constructs TD three-phase data from three sets of two-phase data, provided they satisfy a TD-compatibility condition. The interpolation is formulated as the construction of global capillary pressure and global mobility from two-phase boundary data with Dirichlet and Neumann constraints, optionally choosing a harmonic or biharmonic interior continuation (0901.1462).

For the nonstationary Stokes problem, interpolation pressure arises as post-processing in time. The variational method uses a globally continuous discrete velocity trajectory but allows the discrete pressure trajectory to be discontinuous at time-interval endpoints. The key observation is that the pressure at interval midpoints $(T,\rho)$8 is uniquely defined and optimally accurate. The interpolation-based post-processing constructs, for $(T,\rho)$9, a linear function on $\Omega$0 satisfying

$\Omega$1

with the first interval obtained by extrapolation, or equivalently by the local operator

$\Omega$2

The resulting post-processed pressure converges with optimal second order in time and optimal order in space in the $\Omega$3-norm (Anselmann et al., 11 May 2025).

At immersed interfaces, pressure interpolation becomes pressure-jump reconstruction. For a viscous incompressible fluid with forces concentrated on a discrete surface, the pressure jump is determined by the normal component of the boundary force. Earlier C$\Omega$4 triangulated-surface IIM formulations used piecewise-constant normals, which were found to struggle to accurately capture pressure loads. Two replacements were proposed: an $\Omega$5 projection of the discontinuous normal field into a continuous FE space, and a vertex-normal construction using inverse centroid-distance weighting followed by linear interpolation across each element. Numerical experiments showed that jump conditions computed with reconstructed continuous normals reduced leakage by up to six orders of magnitude across a range of pressures (Facci et al., 7 Mar 2026).

4. Equation-of-state interpolation in compact-star and astrophysical matter

In compact-star matter, interpolation pressure often means interpolation between distinct EoSs. The Mixing Interpolation Method (MIM) defines the mixed-phase pressure as

$\Omega$6

where $\Omega$7 and $\Omega$8 are trigonometric switch functions on the finite interval $\Omega$9, and $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$0 is a pressure increment with amplitude $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$1. The method enforces thermodynamic stability, continuity of higher derivatives at the boundaries, and uses $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$2 as the single free parameter controlling the width and pressure excess of the mixed phase. Setting $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$3 recovers the Maxwell construction. The resulting EoS exhibits an intermediate stiffening and a local maximum in the speed of sound for nonzero $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$4; simulations reported that high-mass twin solutions survive for moderate $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$5 and disappear only when the mixed phase becomes very wide, $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$6 of $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$7 (Blaschke et al., 2018).

A second compact-star construction connects crust and core by interpolation in the pressure–baryonic chemical potential plane. For cold neutron stars, the interpolant is

$$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$8

with continuity and monotonicity constraints chosen so that densities are continuous at the core side and a single density discontinuity is allowed at the crust side. The method is thermodynamically and causally consistent, in contrast with direct classical gluing. It affects low-mass-neutron-star observables in particular: the paper reports visible radius differences, $$\min_{\bm d\in\mathbb C^L}\|\bm G\bm d-\bm u^{\mathrm{des}}\|^2+\eta\|\bm d\|^2, \qquad \hat{\bm d}=(\bm G^H\bm G+\eta\bm I)^{-1}\bm G^H\bm u^{\mathrm{des}},$$9-mode frequency shifts up to about $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$0 at $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$1, minor changes in $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$2 damping times, and no significant effect on the fundamental $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$3-mode or maximum mass (Canullan-Pascual et al., 4 Feb 2025).

Interpolation pressure also appears in tabulated astrophysical EoSs as interpolation of pressure together with its derivatives. Quintic Hermite 2D-splines interpolate $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$4 as a function of temperature and density while preserving mathematical relations between derivatives and therefore thermodynamic identities even between mesh points. The method requires pressure and its first and second derivatives at every mesh point; missing derivatives are recovered from thermodynamic equalities or from differentiated B-splines. On the SAHA-S grid, the maximum $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$5 difference relative to cubic B-spline interpolation lies between $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$6 and $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$7, while the $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$8 difference lies between $$J=\int_\Omega |u_{\mathrm{syn}}(\bm r)-u_{\mathrm{des}}(\bm r)|^2\,d\bm r,$$9 and $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$0. The poorest precision occurs in the dissociation and ionization regions, $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$1, and the best precision at $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$2 (Baturin et al., 2019).

5. Measurement-driven, planetary, and surrogate-model pressure interpolation

In data-driven pressure rendering on a finger pad, interpolation pressure is literal interpolation of measured sensor values. Pressure distributions were collected on a $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$3 capacitance-type sensor array for contact angles $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$4, $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$5, and $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$6, and for pushing displacements in $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$7 steps up to $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$8. For each sensor element, bilinear interpolation is applied in displacement $$J\approx (\bm G\bm d-\bm u^{\mathrm{des}})^H\bm W(\bm G\bm d-\bm u^{\mathrm{des}}), \qquad \hat{\bm d}=(\bm G^H\bm W\bm G+\eta\bm I)^{-1}\bm G^H\bm W\bm u^{\mathrm{des}}.$$9 and contact angle $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$0: $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$1

$\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$2

The Leap Motion Controller supplies displacement and angle in real time. The prediction reproduces the measured data at sample points, and the full distribution over all 240 channels is computed in less than $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$3 (Sase et al., 2024).

On Mars, pressure interpolation was developed to compare Viking 1 and InSight measurements recorded 40 years apart. For nearby sites, the local method assumes hydrostatic equilibrium,

$\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$4

with the temperature at $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$5 above ground taken from atmospheric models. The reported relative interpolation error is about $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$6 at $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$7. For distant comparisons, a global interpolation adds GCM-based dynamical correction and terrain adjustment,

$\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$8

yielding a total interpolation uncertainty of about $\kappa(\bm r_1,\bm r_2)=j_0(k\|\bm r_1-\bm r_2\|),$9 at $(2\,\mathrm m\times 2\,\mathrm m)$0. With these procedures, the comparison did not show changes larger than $(2\,\mathrm m\times 2\,\mathrm m)$1 in the CO$(2\,\mathrm m\times 2\,\mathrm m)$2 cycle (Lange et al., 2022).

In hybrid surrogate modeling of wind flows, pressure interpolation is embedded in a Tucker–Fourier–NN pipeline. Pressure is one variable in a third-order tensor

$(2\,\mathrm m\times 2\,\mathrm m)$3

with Fourier interpolation of the angular factor and $(2\,\mathrm m\times 2\,\mathrm m)$4-nearest-neighbor Gaussian-kernel interpolation of spatial factors. A neural residual correction is then used to suppress ringing artifacts from Fourier interpolation while preserving physical consistency. At unseen angles $(2\,\mathrm m\times 2\,\mathrm m)$5 and $(2\,\mathrm m\times 2\,\mathrm m)$6, the reported pressure $(2\,\mathrm m\times 2\,\mathrm m)$7 values were $(2\,\mathrm m\times 2\,\mathrm m)$8 and $(2\,\mathrm m\times 2\,\mathrm m)$9 for a pure NN, $(1\,\mathrm m\times 1\,\mathrm m)$0 and $(1\,\mathrm m\times 1\,\mathrm m)$1 for Tucker+Fourier, and $(1\,\mathrm m\times 1\,\mathrm m)$2 and $(1\,\mathrm m\times 1\,\mathrm m)$3 for the hybrid Tucker-NN model; the hybrid was described as about $(1\,\mathrm m\times 1\,\mathrm m)$4 faster to train and using more than $(1\,\mathrm m\times 1\,\mathrm m)$5 fewer parameters than the pure NN benchmark (Shaa et al., 28 Oct 2025).

6. Related analytical uses of pressure interpolation and pressure-gradient correction

In the nonlinear theory of clamped circular plates under arbitrary uniform pressure, the interpolation iterative method is an iterative construction for the Von Karman integral equations. It introduces an interpolation parameter $(1\,\mathrm m\times 1\,\mathrm m)$6,

$(1\,\mathrm m\times 1\,\mathrm m)$7

with $(1\,\mathrm m\times 1\,\mathrm m)$8 obtained from a second integral equation. The Homotopy Analysis Method (HAM) generalizes this structure through convergence-control parameters $(1\,\mathrm m\times 1\,\mathrm m)$9 and $\mu$00, and the interpolation iterative method is recovered when $\mu$01 and $\mu$02. For given central deflection, the pressure itself is expanded as

$\mu$03

The paper reports that HAM-based iteration converges much faster than the interpolation iterative method; for example, at $\mu$04, convergence to very high accuracy was reached with 5th-order HAM iteration in about 100 iterations (Zhong et al., 2016).

In wall-bounded turbulence, pressure enters interpolation not as a directly interpolated scalar field but as a universal correction to profile interpolation. The mean velocity in the overlap layer is written with a pressure-gradient term,

$\mu$05

or equivalently

$\mu$06

with $\mu$07 for pipe flow, $\mu$08 for plane-duct/channel flow, and $\mu$09 for Couette flow. The composite interpolation becomes

$\mu$10

where $\mu$11 is a geometry-specific corner-defect function. This use of interpolation is analytically different from pressure-field reconstruction, but it shows that pressure effects can enter interpolation formulas as asymptotic corrections rather than as interpolated observables (Luchini, 2017).

Taken together, these formulations indicate that interpolation pressure is best understood as a methodological role played by pressure in reconstruction, matching, continuation, or correction. In some settings the object interpolated is the pressure field itself; in others it is a pressure trajectory, a pressure jump, a pressure-dependent state relation, or a pressure-gradient contribution to another interpolant. The unifying principle is that interpolation is used to recover physically constrained behavior between discrete samples, phases, or computational locations.