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

TL;DR

This paper introduces a positivity-preserving kinetic scheme for multi-component Euler equations that ensures accurate and stable solutions across different dimensions and flow features, including shock interactions.

Contribution

The paper presents a novel kinetic model with velocity formulations that preserve positivity and achieve high-order accuracy for multi-component flows.

Findings

Successfully captures shock-bubble interactions.

Maintains positivity of density and pressure.

Achieves third-order accuracy with flux limiting.

Abstract

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.