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Conservative, pressure-equilibrium-preserving discontinuous Galerkin method for compressible, multicomponent flows

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

TL;DR

This work analyzes velocity- and pressure-equilibrium preservation in discontinuous Galerkin discretizations of compressible, multicomponent Euler equations. It shows that a standard DG discretization of the conservative form is velocity-equilibrium-preserving but not pressure-equilibrium-preserving, motivating a pressure-based formulation that replaces total energy with pressure evolution and uses Abgrall-type correction terms to recover semidiscrete energy conservation. However, these corrections initially disrupt equilibria and zero-species preservation, leading to modified correction terms and face-based corrections that restore velocity and pressure equilibria while maintaining energy conservation in a semidiscrete sense. The proposed approach is validated on smooth interfacial flows in 1D–3D (including curved elements), showing reduced spurious pressure oscillations and robust stability compared to total-energy-based schemes, with ongoing work on discontinuities, fully discrete conservation, and real-fluid mixtures.

Abstract

This paper concerns preservation of velocity and pressure equilibria in smooth, compressible, multicomponent flows in the inviscid limit. First, we derive the velocity-equilibrium and pressure-equilibrium conditions of a standard discontinuous Galerkin method that discretizes the conservative form of the compressible, multicomponent Euler equations. We show that under certain constraints on the numerical flux, the scheme is velocity-equilibrium-preserving. However, standard discontinuous Galerkin schemes are not pressure-equilibrium-preserving. Therefore, we introduce a discontinuous Galerkin method that discretizes the pressure-evolution equation in place of the total-energy conservation equation. Semidiscrete conservation of total energy, which would otherwise be lost, is restored via the correction terms of [Abgrall, J. Comput. Phys., 372, 2018, pp. 640-666] and [Abgrall et al., J. Comput. Phys., 453, 2022, 110955]. Since the addition of the correction terms prevents exact preservation of pressure and velocity equilibria, we propose modifications that then lead to a velocity-equilibrium-preserving, pressure-equilibrium-preserving, and (semidiscretely) energy-conservative discontinuous Galerkin scheme, although there are certain tradeoffs. Additional extensions are also introduced. We apply the developed scheme to smooth, interfacial flows involving mixtures of thermally perfect gases initially in pressure and velocity equilibria to demonstrate its performance in one, two, and three spatial dimensions.

Conservative, pressure-equilibrium-preserving discontinuous Galerkin method for compressible, multicomponent flows

TL;DR

This work analyzes velocity- and pressure-equilibrium preservation in discontinuous Galerkin discretizations of compressible, multicomponent Euler equations. It shows that a standard DG discretization of the conservative form is velocity-equilibrium-preserving but not pressure-equilibrium-preserving, motivating a pressure-based formulation that replaces total energy with pressure evolution and uses Abgrall-type correction terms to recover semidiscrete energy conservation. However, these corrections initially disrupt equilibria and zero-species preservation, leading to modified correction terms and face-based corrections that restore velocity and pressure equilibria while maintaining energy conservation in a semidiscrete sense. The proposed approach is validated on smooth interfacial flows in 1D–3D (including curved elements), showing reduced spurious pressure oscillations and robust stability compared to total-energy-based schemes, with ongoing work on discontinuities, fully discrete conservation, and real-fluid mixtures.

Abstract

This paper concerns preservation of velocity and pressure equilibria in smooth, compressible, multicomponent flows in the inviscid limit. First, we derive the velocity-equilibrium and pressure-equilibrium conditions of a standard discontinuous Galerkin method that discretizes the conservative form of the compressible, multicomponent Euler equations. We show that under certain constraints on the numerical flux, the scheme is velocity-equilibrium-preserving. However, standard discontinuous Galerkin schemes are not pressure-equilibrium-preserving. Therefore, we introduce a discontinuous Galerkin method that discretizes the pressure-evolution equation in place of the total-energy conservation equation. Semidiscrete conservation of total energy, which would otherwise be lost, is restored via the correction terms of [Abgrall, J. Comput. Phys., 372, 2018, pp. 640-666] and [Abgrall et al., J. Comput. Phys., 453, 2022, 110955]. Since the addition of the correction terms prevents exact preservation of pressure and velocity equilibria, we propose modifications that then lead to a velocity-equilibrium-preserving, pressure-equilibrium-preserving, and (semidiscretely) energy-conservative discontinuous Galerkin scheme, although there are certain tradeoffs. Additional extensions are also introduced. We apply the developed scheme to smooth, interfacial flows involving mixtures of thermally perfect gases initially in pressure and velocity equilibria to demonstrate its performance in one, two, and three spatial dimensions.
Paper Structure (22 sections, 6 theorems, 100 equations, 17 figures)

This paper contains 22 sections, 6 theorems, 100 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Compressible, multicomponent Euler equations: Total-energy-based formulation
  3. Discontinuous Galerkin discretization
  4. Velocity-equilibrium preservation
  5. Pressure-equilibrium preservation
  6. Compressible, multicomponent Euler equations: Pressure-based formulation
  7. Discontinuous Galerkin discretization
  8. Pressure-equilibrium preservation
  9. Correction term: Original formulation
  10. Preservation of velocity and pressure equilibria
  11. Correction term: Modified formulation
  12. Preservation of velocity and pressure equilibria
  13. Results
  14. Gaussian density wave
  15. One-dimensional, high-velocity thermal bubble
  16. ...and 7 more sections

Key Result

Lemma 1

In the case of constant pressure $(P=P_{0})$ and velocity $\left(\bm{v}=\bm{v}_{0}\right)$, the DG discretization (eq:dg-integral-form-conservative) preserves velocity equilibrium (i.e., $\partial_{t}v=0$ necessarily) if and only if the numerical flux satisfies where $\mathcal{F}_{\rho v}^{\dagger}\left(\bm{y}^{+},\bm{y}^{-},n\right)$ is the momentum component of the numerical flux and $\mathcal{

Figures (17)

  • Figure 4.1: Convergence under grid refinement, with $h=0.02$, for the multicomponent Gaussian-wave test. The $L^{2}$ error of the normalized state with respect to the exact solution at $t=1$ is computed. The dashed lines represent convergence rates of $p+1$.
  • Figure 4.2: Convergence of total-energy conservation with respect to time-step size, where $\Delta t=3.14\text{ $\mu s$}$, for the one-dimensional, high-velocity thermal-bubble test case. The percent error in global energy conservation after 100 advection periods is computed. The dashed line represents a third-order rate of convergence.
  • Figure 4.3: $p=3$ solutions to the advection of a nitrogen/n-dodecane thermal bubble at $v=600\text{ m/s}$. P1: uncorrected pressure--based DG scheme; P2: pressure-based DG scheme with original correction term (\ref{['eq:correction-term-original']}); P3: proposed pressure-based DG scheme (Section \ref{['subsec:correction-term-modified']}); E1: total-energy-based DG scheme with overintegration; E2: total-energy-based DG scheme with colocated integration; Exact: exact solution.
  • Figure 4.4: $Y_{\mathrm{O_{2}}}$ profiles at $t=100\tau$ obtained from $p=3$ solutions to the advection of a nitrogen/n-dodecane thermal bubble at $v=600\text{ m/s}$.
  • Figure 4.5: $p=3$ solution to the advection of a nitrogen/n-dodecane thermal bubble at $v=1\text{ m/s}$. P1: uncorrected pressure--based DG scheme; P2: pressure-based DG scheme with original correction term (\ref{['eq:correction-term-original']}); P3: proposed pressure-based DG scheme (Section \ref{['subsec:correction-term-modified']}); E1: total-energy-based DG scheme with overintegration; E2: total-energy-based DG scheme with colocated integration; Exact: exact solution.
  • ...and 12 more figures

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 6
  • proof
  • Remark 7
  • Remark 8
  • ...and 15 more