On the Construction of High-Order and Exact Pressure Equilibrium Schemes for Arbitrary Equations of State

TL;DR

<3-5 sentence high-level summary> Spurious pressure oscillations in conservative multispecies Euler simulations with real equations of state motivate the development of pressure-equilibrium preserving schemes. The authors present two complementary approaches: a high-order approximate pressure-equilibrium preserving (APEC) scheme and a fully conservative, spatially exact pressure-equilibrium preserving (PEP) scheme, both applicable to arbitrary EOS and any number of species. The high-order APEC extension distributes thermodynamic corrections across sub-interfaces to substantially reduce pressure oscillations, while the fully conservative PEP method enforces exact PEP via interface corrections (α,β) using a robust Moore–Penrose solve with a conditioning-based global threshold. Numerical experiments on ideal-gas, stiffened-gas, and van der Waals mixtures demonstrate dramatic reductions in spurious pressure oscillations and quantify the trade-offs in accuracy, robustness, and long-time behavior, outlining pathways for hybridization and further high-order development.

Abstract

Typical fully conservative discretizations of the Euler compressible single or multi-component fluid equations governed by a real-fluid equation of state exhibit spurious pressure oscillations due to the nonlinearity of the thermodynamic relation between pressure, density, and internal energy. A fully conservative, pressure-equilibrium preserving method and a high-order, fully conservative, approximate pressure-equilibrium preserving method are presented. Both methods are general and can handle an arbitrary equation of state and arbitrary number of species. Unlike existing approaches to discretize the multi-component Euler equations, we do not introduce non conservative updates, overspecified equations, or design for a specific equation of state. The proposed methods are demonstrated on inviscid smooth interface advection problems governed by three equations of state: ideal-gas, stiffened-gas, and van der Waals where we show orders of magnitude reductions in spurious pressure oscillations compared to existing schemes.

Figures (13)

• Figure 1: Construction of the high-order interface flux at $i+1/2$ as a weighted sum of multiple cell-pair contributions. Dashed gray lines connect cell centers to their arrow origins. For second-order accuracy (blue), a single pair $\langle i, i+1 \rangle$ forms the sub-interface state. Fourth-order accuracy adds two additional pairs: $\langle i-1, i+1 \rangle$ (orange) and $\langle i, i+2 \rangle$ (green). Sixth-order accuracy further extends the stencil with three more pairs: $\langle i-2, i+1 \rangle$ (red), $\langle i-1, i+2 \rangle$ (purple), and $\langle i, i+3 \rangle$ (brown). Each arrow pair represents a sub-interface contribution formed by averaging the pointwise values at two cell centers, and the final interface flux is the weighted sum of all these sub-interface contributions according to Equation \ref{['eq:high_order_flux']}.
• Figure 2: Second-order numerical solutions at $t=20$ for the ideal-gas test case defined in Equation \ref{['eq:ideal_gas_ICs']}. The four panels show: (top-left) the proposed PEP scheme; (top-right) the APEC scheme of TERASHIMA2025113701; (bottom-left) the PEP method of fujiwara_fully_2023; and (bottom-right) the second-order KEEP scheme (FC-NPE). In each panel, blue, orange, green, and red denote $\rho Y_1$, $\rho Y_2$, $u$, and $p$, respectively.
• Figure 3: $L_2$ pressure error (left) and total–energy error (right) for the ideal-gas test case. Curves correspond to the APEC $\mathcal{O}(2)$ scheme (blue), the FC-NPE scheme (orange), the Fujiwara PEP scheme (green), and the proposed PEP scheme (red).
• Figure 4: $L_2$ pressure error (left) and total–energy error (right) for the higher–order APEC and NPE schemes applied to the ideal-gas test. APEC curves are plotted as solid lines and NPE curves as dashed lines, with colors denoting the formal order of accuracy: $\mathcal{O}(2)$ (blue), $\mathcal{O}(4)$ (orange), $\mathcal{O}(6)$ (green), and $\mathcal{O}(8)$ (red). The proposed PEP scheme is shown in purple for reference.
• Figure 5: Second-order numerical solutions at $t=7$ for the stiffened-gas test case defined in Equation \ref{['eq:ideal_gas_ICs']}. The four panels show: (top-left) the proposed PEP scheme; (top-right) the APEC scheme of TERASHIMA2025113701; (bottom-left) the PEP method of fujiwara_fully_2023; and (bottom-right) the second-order KEEP scheme (FC-NPE). In each panel, blue, orange, green, and red denote $\rho Y_1$, $\rho Y_2$, $u$, and $p$, respectively.