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Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Eric J. Ching, Ryan F. Johnson, Sarah Burrows, Jacklyn Higgs, Andrew D. Kercher

TL;DR

The authors develop a fully conservative, positivity-preserving, and entropy-bounded DG method for the chemically reacting, multicomponent, compressible Navier–Stokes equations, extending prior inviscid results to viscous flows. A separable positivity-preserving Lax–Friedrichs-type viscous flux is adapted to multicomponent diffusion, with entropy bounding applied to the convective part while ensuring mass conservation and compatibility with curved elements. The framework includes a rigorous limiting strategy, boundary-condition treatments, and an adaptive time-stepping procedure that balances BR2 and LF flux usage to maximize robustness and efficiency. Demonstrations on 1D, 2D, and 3D test problems—ranging from thermal bubbles to detonation waves and shock/mixing-layer interactions—highlight optimal convergence for smooth flows and improved stability in reacting flows through entropy bounding. The results support the method’s potential for robust, high-fidelity simulations of complex, chemically reacting flows on high-order, curved meshes, with future extensions to larger-scale geometries and richer chemistries.

Abstract

This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also detail how to account for boundary conditions and incorporate previously developed techniques to reduce spurious pressure oscillations into the positivity-preserving framework. Furthermore, potential issues associated with the Lax-Friedrichs-type viscous flux function in the case of zero species concentrations are discussed and addressed. The resulting formulation is compatible with curved, multidimensional elements and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction.

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

TL;DR

The authors develop a fully conservative, positivity-preserving, and entropy-bounded DG method for the chemically reacting, multicomponent, compressible Navier–Stokes equations, extending prior inviscid results to viscous flows. A separable positivity-preserving Lax–Friedrichs-type viscous flux is adapted to multicomponent diffusion, with entropy bounding applied to the convective part while ensuring mass conservation and compatibility with curved elements. The framework includes a rigorous limiting strategy, boundary-condition treatments, and an adaptive time-stepping procedure that balances BR2 and LF flux usage to maximize robustness and efficiency. Demonstrations on 1D, 2D, and 3D test problems—ranging from thermal bubbles to detonation waves and shock/mixing-layer interactions—highlight optimal convergence for smooth flows and improved stability in reacting flows through entropy bounding. The results support the method’s potential for robust, high-fidelity simulations of complex, chemically reacting flows on high-order, curved meshes, with future extensions to larger-scale geometries and richer chemistries.

Abstract

This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also detail how to account for boundary conditions and incorporate previously developed techniques to reduce spurious pressure oscillations into the positivity-preserving framework. Furthermore, potential issues associated with the Lax-Friedrichs-type viscous flux function in the case of zero species concentrations are discussed and addressed. The resulting formulation is compatible with curved, multidimensional elements and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction.
Paper Structure (36 sections, 4 theorems, 170 equations, 15 figures, 1 table)

This paper contains 36 sections, 4 theorems, 170 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Discontinuous Galerkin discretization
  4. Transport step: Positivity-preserving, entropy-bounded discontinuous Galerkin method
  5. Positivity-preserving Lax-Friedrichs-type viscous flux function
  6. One-dimensional case
  7. First-order DG scheme in one dimension
  8. High-order DG scheme in one dimension
  9. Modified flux interpolation
  10. Limiting procedure
  11. Multidimensional case
  12. Preliminaries
  13. Geometric mapping
  14. Quadrature rules
  15. Additional considerations
  16. ...and 21 more sections

Key Result

Lemma 1

Assume that $y=\left(\rho v,\rho e_{t},C\right)^{T}$ is in $\mathcal{G}$ and that $C_{i}>0,\:\forall i$. Then $y\pm\beta^{-1}\mathcal{F}^{v}\cdot n$, where $n$ is a given unit vector, is also in $\mathcal{G}$ under the following conditions: where with $b=\rho\mathcal{F}_{\rho e_{t}}^{v}\cdot n-\rho v\cdot\mathcal{F}_{\rho v}^{v}\cdot n$.

Figures (15)

  • Figure 4.1: Schematic of an example scenario in which $\overline{C}_{i,\kappa,v}^{j+1}<0$ regardless of $\beta$.
  • Figure 5.1: Convergence under grid refinement, with $h=2\:\mathrm{m}$, for the one-dimensional thermal bubble test case. The $L^{2}$ error of the normalized state with respect to the exact solution at $t=5\:\mathrm{s}$ is computed. The dashed lines represent the theroretical convergence rates. The "$\times$" symbol indicates that the positivity-preserving limiter is activated, the "$\Circle$" symbol indicates that the entropy limiter is activated, and the "$\triangle$" symbol indicates that neither limiter is activated. If both limiters are activated, then the corresponding symbols are superimposed as "$\otimes$".
  • Figure 5.2: $p=1$ solution to a one-dimensional premixed flame at $t=0.005$ s obtained with species clipping.
  • Figure 5.3: $p=1$ solution to a one-dimensional premixed flame at $t=0.005$ s obtained with the proposed positivity-preserving and entropy-bounded formulation.
  • Figure 5.4: $p=3$ solution to a one-dimensional premixed flame at $t=0.005$ s obtained with the proposed positivity-preserving and entropy-bounded formulation.
  • ...and 10 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • proof
  • Remark 7
  • Theorem 8
  • ...and 8 more