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

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

Abstract

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.

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

Abstract

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.
Paper Structure (26 sections, 2 theorems, 133 equations, 18 figures, 5 tables)

This paper contains 26 sections, 2 theorems, 133 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Summary of Part I
  3. Contributions of Part II
  4. Additional background
  5. Outline
  6. Governing equations
  7. Discontinuous Galerkin discretization
  8. Transport step: Entropy-bounded discontinuous Galerkin method in multiple dimensions
  9. Preliminaries
  10. Geometric mapping
  11. Quadrature rules
  12. Minimum entropy principle
  13. First-order three-point system
  14. Entropy-bounded, high-order discontinuous Galerkin method in multiple dimensions
  15. Limiting procedure
  16. ...and 11 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-2D-unexpanded) is also in $\mathcal{G}_{\sigma}$ under the constraint and the conditions

Figures (18)

  • Figure 4.1: Schematic of a rotated three-point system in the multidimensional setting (Equation (\ref{['eq:three-point-system-2D']})).
  • Figure 6.1: $p=1$ solution to two-dimensional thermal-bubble advection on a curved quadrilateral grid without artificial viscosity. The final time is $t=50$ s, corresponding to one period. The colorbar minima and maxima for the pressure and velocity fields are the respective global minima and maxima.
  • Figure 6.2: $p=2$ solution to two-dimensional thermal-bubble advection on a curved quadrilateral grid without artificial viscosity. The final time is $t=50$ s, corresponding to one period. The colorbar minima and maxima for the pressure and velocity fields are the respective global minima and maxima.
  • Figure 6.3: $p=3$ solution to two-dimensional thermal-bubble advection on a curved quadrilateral grid without artificial viscosity. The final time is $t=50$ s, corresponding to one period. The colorbar minima and maxima for the pressure and velocity fields are the respective global minima and maxima.
  • Figure 6.4: OH mole-fraction field for a two-dimensional moving detonation wave at $t=200$$\mu\mathrm{s}$, computed with $p=2$ on a sequence of meshes, where $h=9\times10^{-5}$ m. The initial conditions are given in Equation (\ref{['eq:2D-detonation-initialization']}).
  • ...and 13 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof
  • Remark 6