Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Second-order invariant-domain preserving approximation to the multi-species Euler equations

Bennett Clayton, Tarik Dzanic, Eric J. Tovar

TL;DR

The paper addresses the challenge of robustly solving the compressible multi-species Euler equations while strictly preserving physical invariants. It derives the full 1D Riemann problem and a maximum wave speed bound, then builds a first-order invariant-domain preserving scheme and extends it to second-order accuracy using a convex limiting approach combined with an entropy indicator. The resulting method preserves positivity of species densities and internal energy, enforces a local minimum principle on mixture entropy, and satisfies discrete entropy inequalities. Numerical experiments, including 1D traveling waves, Riemann problems, 2D shock–bubble interactions, an ICF-like configuration, and a 3D axisymmetric triple-point scenario, demonstrate accuracy, robustness, and the ability to capture complex multi-species mixing and shocks with high fidelity.

Abstract

This work is concerned with constructing a second-order, invariant-domain preserving approximation of the compressible multi-species Euler equations where each species is modeled by an ideal gas equation of state. We give the full solution to the Riemann problem and derive its maximum wave speed. The maximum wave speed is used in constructing a first-order invariant-domain preserving approximation. We then extend the methodology to second-order accuracy and detail a convex limiting technique which is used for preserving the invariant domain. Finally, the numerical method is verified with analytical solutions and then validated with several benchmarks and laboratory experiments.

Second-order invariant-domain preserving approximation to the multi-species Euler equations

TL;DR

The paper addresses the challenge of robustly solving the compressible multi-species Euler equations while strictly preserving physical invariants. It derives the full 1D Riemann problem and a maximum wave speed bound, then builds a first-order invariant-domain preserving scheme and extends it to second-order accuracy using a convex limiting approach combined with an entropy indicator. The resulting method preserves positivity of species densities and internal energy, enforces a local minimum principle on mixture entropy, and satisfies discrete entropy inequalities. Numerical experiments, including 1D traveling waves, Riemann problems, 2D shock–bubble interactions, an ICF-like configuration, and a 3D axisymmetric triple-point scenario, demonstrate accuracy, robustness, and the ability to capture complex multi-species mixing and shocks with high fidelity.

Abstract

This work is concerned with constructing a second-order, invariant-domain preserving approximation of the compressible multi-species Euler equations where each species is modeled by an ideal gas equation of state. We give the full solution to the Riemann problem and derive its maximum wave speed. The maximum wave speed is used in constructing a first-order invariant-domain preserving approximation. We then extend the methodology to second-order accuracy and detail a convex limiting technique which is used for preserving the invariant domain. Finally, the numerical method is verified with analytical solutions and then validated with several benchmarks and laboratory experiments.
Paper Structure (28 sections, 8 theorems, 60 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 28 sections, 8 theorems, 60 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The model problem
  3. Thermodynamics
  4. Model properties
  5. The Riemann problem
  6. Set up and summary
  7. Elementary wave structure
  8. Solution to the Riemann problem
  9. Approximation details
  10. Low-order method
  11. Local bounds
  12. Provisional high-order method
  13. Entropy indicator
  14. Convex limiting
  15. Set up
  16. ...and 13 more sections

Key Result

Proposition 3.1

\newlabelprop:wave_speeds0 Let $Z \in \{L, R\}$. Assume $p^{*}$ is a solution to $\varphi(p)=0$ where and $A_Z = \frac{2}{(\gamma_Z + 1)\rho_Z}$, $B_Z = \frac{\gamma_Z - 1}{\gamma_Z + 1} p_Z$, $c_Z = \sqrt{\frac{\gamma_Z p_Z}{\rho_Z}}$. Then the maximum wave speed is given by where

Figures (8)

  • Figure 1: Entropy indicator illustration with multi-species Woodward-Colella blast wave.
  • Figure 2: Mixture density (left) and pressure (right) at $t_f$ for the 1D Riemann problems computed using varying mesh resolutions. \newlabelfig:rp0
  • Figure 3: 2D Shock-bubble -- Numerical schlieren output (with respect to the air partial density) at $t = \{0, 246, 492, 738, 984, 1230\}\mu s$.
  • Figure 4: 2D Shock-bubble -- A zoomed in snapshot of the mass fraction for krypton at $t = {\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{1230}{\mu s}}$.
  • Figure 5: 2D ICF-like problem -- Initial set up.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Remark 2.1: Alternative formulation
  • Remark 2.2: Mixture adiabatic index
  • Remark 2.3: Dalton's Law and material quantities
  • Definition 2.4: Entropy solutions
  • Definition 2.5: Invariant set
  • Proposition 3.1: Maximum wave speed
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3: Mass fractions
  • Proof 2
  • ...and 10 more