A Kinetic Scheme Based On Positivity Preservation For Multi-component Euler Equations

A kinetic model with flexible velocities is presented for solving the multi-component Euler equations. The model employs a two-velocity formulation in 1D and a three-velocity formulation in 2D. In 2D, the velocities are aligned with the cell-interface to ensure a locally one-dimensional macroscopic normal flux in a finite volume. The velocity magnitudes are defined to satisfy conditions for preservation of positivity of density of each component as well as of overall pressure for first order accuracy under a CFL-like time-step restriction. Additionally, at a stationary contact discontinuity, the velocity definition is modified to achieve exact capture. The basic scheme is extended to third order accuracy using a Chakravarthy-Osher type flux-limited approach along with Strong Stability Preserving Runge-Kutta (SSPRK) method. Benchmark 1D and 2D test cases, including shock-bubble interaction problems, are solved to demonstrate the efficacy of the solver in accurately capturing the relevant flow features.

💡 Research Summary

The paper introduces a novel kinetic scheme tailored for the multi‑component Euler equations, addressing longstanding challenges in preserving physical admissibility and accurately capturing material interfaces. The authors construct a vector‑kinetic framework that replaces the traditional Maxwellian equilibrium with a pair of Dirac‑delta functions positioned at velocities +λ and –λ for each species in one dimension, and a three‑velocity set (λ, 0, –λ) aligned with cell faces in two dimensions. By defining λ as a flexible parameter subject to analytically derived lower bounds—functions of the local flow velocity, sound speed, and specific‑heat ratios—the scheme guarantees positivity of every component’s density and of the total pressure under a CFL‑type time‑step restriction.

A key innovation is the modification of λ at stationary contact discontinuities: λ is set equal to the local fluid velocity, ensuring that the numerical flux exactly reproduces the physically stationary material interface without generating spurious pressure oscillations—a common deficiency of conventional conservative schemes. This exact capture of steady contacts is demonstrated analytically and verified numerically.

The basic first‑order explicit method is then elevated to third‑order accuracy through a Chakravarthy‑Osher flux‑limiter combined with a Strong Stability Preserving Runge‑Kutta (SSP‑RK) time integrator. The limiter suppresses non‑physical overshoots while preserving total variation diminishing (TVD) properties, and the SSP‑RK scheme maintains the positivity constraints at higher order.

Extensive benchmark problems validate the approach. One‑dimensional tests include Sod and Lax shock tubes, contact discontinuities, and multi‑species wave interactions, all of which confirm that component densities and pressure remain positive and that contact waves are sharply resolved. Two‑dimensional experiments cover shock‑bubble interactions, multi‑component shear‑layer problems, and complex wave‑structure interactions. In the shock‑bubble case, the scheme accurately tracks bubble deformation, vortex shedding, and transmitted shocks without any loss of positivity or emergence of pressure anomalies, outperforming many existing kinetic and finite‑volume methods.

Theoretical analysis is provided in an appendix, where the positivity conditions are rigorously derived, and the CFL constraint for the third‑order scheme is linked to the eigenvalue bounds of the underlying kinetic operator.

Overall, the work delivers a robust, eigen‑structure‑independent kinetic solver that simultaneously ensures (i) component‑wise density positivity, (ii) overall pressure positivity, (iii) exact stationary contact resolution, and (iv) high‑order accuracy with strong stability. These attributes make the method a compelling candidate for realistic multi‑component compressible flow simulations where traditional schemes struggle with robustness and interface fidelity.