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Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case

Eric J. Ching, Ryan F. Johnson, Andrew D. Kercher

TL;DR

The enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case.

Abstract

In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the conservative, high-order numerical method previously developed by Johnson and Kercher [J. Comput. Phys., 423 (2020), 109826] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. [ESAIM: Math. Model. Numer. Anal., 54 (2020), 373--389] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients required for the solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case.

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case

TL;DR

The enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case.

Abstract

In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the conservative, high-order numerical method previously developed by Johnson and Kercher [J. Comput. Phys., 423 (2020), 109826] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. [ESAIM: Math. Model. Numer. Anal., 54 (2020), 373--389] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients required for the solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case.
Paper Structure (32 sections, 6 theorems, 182 equations, 9 figures)

This paper contains 32 sections, 6 theorems, 182 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Governing equations
  4. Chemical reaction rates
  5. Arrhenius reactions
  6. Three-body reactions
  7. Unimolecular/recombination fall-off reactions
  8. Chemically activated bimolecular reactions
  9. Discontinuous Galerkin discretization
  10. Minimum entropy principle
  11. Review: Minimum entropy principle in the compressible, multicomponent, nonreacting Euler equations
  12. Minimum entropy principle in the compressible, multicomponent, reacting Euler equations
  13. Transport step: Entropy-bounded discontinuous Galerkin scheme in one dimension
  14. Preliminaries
  15. Entropy-bounded, high-order discontinuous Galerkin method in one dimension
  16. ...and 17 more sections

Key Result

Theorem 1

If $y_{\kappa}^{j}(x)\in\mathcal{G}_{\sigma},\;\forall x\in\mathcal{D_{\kappa}}$, and $y_{\kappa}^{-,j}\in\mathcal{G}_{\sigma},\;\forall x\in\partial\mathcal{D}_{\kappa}$, with then $\overline{y}_{\kappa}^{j+1}$ in Equation (eq:fully-discrete-form-average-unexpanded) is also in $\mathcal{G}_{\sigma}$ under the constraint and the conditions

Figures (9)

  • Figure 7.1: 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 "$\times$" marker indicates that the positivity-preserving limiter is activated, the "$\Circle$" marker indicates that the entropy limiter is activated, and the "$\triangle$" marker indicates that neither limiter is activated. If both limiters are activated, then the corresponding markers are superimposed as "$\otimes$".
  • Figure 7.2: Temperature profiles at $t=50\:s$ computed with the local entropy bound in Lv15_2 ("Old") and the local entropy bound in Equation (\ref{['eq:local-entropy-bound']}) ("New"). The exact solution is the same as the initial condition. The element size is $h/4$ and the polynomial order is $p=1$.
  • Figure 7.3: Results for $p=3$ solutions on 200 elements without artificial viscosity for the one-dimensional, multicomponent shock-tube problem with initialization in Equation (\ref{['eq:shock-tube-IC-Houim']}). "PPL" corresponds to the positivity-preserving limiter by itself, and "local EB" refers to both the positivity-preserving and entropy limiters with the local entropy bound in Equation (\ref{['eq:local-entropy-bound']}). The reference solution Joh20_2 is computed with $p=2$, 2000 elements, and artificial viscosity.
  • Figure 7.4: Percent error in conservation of mass, energy, and atomic elements for the "local EB" case in Figure \ref{['fig:shock_tube_Houim']}, computed with $p=3$ on 200 elements. The initial conditions for this one-dimensional, multicomponent shock-tube problem are given in Equation (\ref{['eq:shock-tube-IC-Houim']}). Also included is the error in mass conservation for a solution computed without the positivity-preserving and entropy limiters, but instead with a simple clipping procedure in which negative species concentrations are set to zero.
  • Figure 7.5: Results for $p=3$ solutions on 200 elements with artificial viscosity for the one-dimensional, multicomponent shock-tube problem with initialization in Equation (\ref{['eq:shock-tube-IC-Houim']}). "PPL" corresponds to the positivity-preserving limiter by itself, and "local EB" refers to both the positivity-preserving and entropy limiters with the local entropy bound in Equation (\ref{['eq:local-entropy-bound']}). The reference solution Joh20_2 is computed with $p=2$, 2000 elements, and artificial viscosity. The difference between these results and those in Figure \ref{['fig:shock_tube_Houim']} is the use of artificial viscosity in the (non-reference) solutions here.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Theorem 1: Zha10Zha12_2Lv15_2
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Remark 7
  • ...and 7 more