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A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

Rémi Abgrall, Pratik Rai, Florent Renac

TL;DR

The paper develops a high-order DGSEM framework for the nonconservative SG-gamma multicomponent model on unstructured curved meshes, combining entropy-stable EC fluctuations in the volume with an HLLC Riemann solver at interfaces. It demonstrates semi-discrete entropy stability (away from material fronts) and material-interface preservation (with CP fluctuations), provides rigorous wave-speed estimates and discrete entropy/positivity results, and confirms high-order accuracy via extensive 1D and 2D tests including density-wave advection and shock–bubble interactions. The key contribution is a robust, root-free HLLC-based interface treatment that accommodates nonconservative terms while preserving essential physical properties such as pressure equilibrium, positivity, and entropy structure. The work offers a practical, scalable approach for simulating complex multiphase/multicomponent flows with curved high-order meshes, with potential impact on simulations of shocks, interfaces, and phase interactions in engineering and physics.

Abstract

In this work, we propose an accurate, robust, and stable discretization of the gamma-based compressible multicomponent model by Shyue [J. Comput. Phys., 142 (1998), 208-242] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac [J. Comput. Phys. 382 (2019), 1-26] and Coquel et al. [J. Comput. Phys. 431 (2021) 110135] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on the discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization and prove that the scheme is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver that does not require a root-finding algorithm for approximating the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, preserves uniform states across material interfaces, satisfies a discrete minimum principle on the specific entropy and maximum principles on the EOS parameters. Third, the HLLC solver is applied at interfaces in the DGSEM scheme, while we consider two kinds of fluctuations in the integrals over discretization elements: material interface preserving and entropy conservative. Time integration is performed using SSP Runge-Kutta schemes. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

TL;DR

The paper develops a high-order DGSEM framework for the nonconservative SG-gamma multicomponent model on unstructured curved meshes, combining entropy-stable EC fluctuations in the volume with an HLLC Riemann solver at interfaces. It demonstrates semi-discrete entropy stability (away from material fronts) and material-interface preservation (with CP fluctuations), provides rigorous wave-speed estimates and discrete entropy/positivity results, and confirms high-order accuracy via extensive 1D and 2D tests including density-wave advection and shock–bubble interactions. The key contribution is a robust, root-free HLLC-based interface treatment that accommodates nonconservative terms while preserving essential physical properties such as pressure equilibrium, positivity, and entropy structure. The work offers a practical, scalable approach for simulating complex multiphase/multicomponent flows with curved high-order meshes, with potential impact on simulations of shocks, interfaces, and phase interactions in engineering and physics.

Abstract

In this work, we propose an accurate, robust, and stable discretization of the gamma-based compressible multicomponent model by Shyue [J. Comput. Phys., 142 (1998), 208-242] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac [J. Comput. Phys. 382 (2019), 1-26] and Coquel et al. [J. Comput. Phys. 431 (2021) 110135] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on the discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization and prove that the scheme is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver that does not require a root-finding algorithm for approximating the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, preserves uniform states across material interfaces, satisfies a discrete minimum principle on the specific entropy and maximum principles on the EOS parameters. Third, the HLLC solver is applied at interfaces in the DGSEM scheme, while we consider two kinds of fluctuations in the integrals over discretization elements: material interface preserving and entropy conservative. Time integration is performed using SSP Runge-Kutta schemes. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.
Paper Structure (28 sections, 7 theorems, 137 equations, 11 figures, 1 table)

This paper contains 28 sections, 7 theorems, 137 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The SG-gamma model
  3. Governing equations and thermodynamic model
  4. Entropy pair
  5. The discontinuous Galerkin spectral element method (DGSEM)
  6. Numerical approximation and function space
  7. Semi-discrete form
  8. Numerical fluctuations for the volume integrals
  9. Contact preserving numerical fluxes
  10. Entropy conservative numerical fluxes
  11. An HLLC Riemann solver for the cell interfaces
  12. One-dimensional Riemann problem
  13. Three-point schemes and the Godunov method
  14. HLLC Riemann solver
  15. Properties of the HLLC solver
  16. ...and 13 more sections

Key Result

Theorem 3.1

Let $\normalfont{\textbf{D}}^\pm_{X}(\cdot,\cdot,\cdot)$ in (eqn: chap 5 EC fluxes abstract form) be consistent fluctuations with ${\bf d}_{X}^\pm(\cdot,\cdot,\cdot)$ in (eqn: chap 3 tilde D) satisfying where $\normalfont{\textbf{n}}=(n_1,\dots,n_d)^\top$, ${\bf c}(\normalfont{\textbf{u}}){\bf n}=\sum_{i=1}^dn_i\textbf{c}_i(\normalfont{\textbf{u}})$, and $\llbracket \normalfont{\textbf{u}}\rrbrac

Figures (11)

  • Figure 1: Notations for the mesh in two space dimensions ($d=2$): cell $\kappa\in\Omega_h$ with quadrature points $\normalfont{\textbf{x}}_\kappa^{ij}$ (bullets $\bullet$), edge $e\in\partial\kappa$ with quadrature point $\normalfont{\textbf{x}}_e^k$, associated unit outward normal $\normalfont{\textbf{n}}_e^k$, traces of the approximate solution $\normalfont{\textbf{u}}^\pm_h$ on $e$; and adjacent cell $\kappa_e^+$ sharing edge $e$.
  • Figure 2: The above representation shows two cells, where the left cell contains a material interface (line in color) which separates the states $\Omega_L = \{\alpha_{1L},\rho_L,\Gamma_L,\pi_L\}$ and $\Omega_R = \{\alpha_{1R},\rho_R,\Gamma_R,\pi_R\}$ with discontinuous density and EOS parameters and uniform velocity $\normalfont{\textbf{v}}$ and pressure $\mathrm{p}$. Note that the present scheme does not require the material discontinuity to be aligned with mesh interfaces to preserve uniform pressure and velocity profiles across it.
  • Figure 3: Notations for the mesh used for the three-point scheme (\ref{['eq:3pt-scheme-NC']}).
  • Figure 4: Wave pattern of the HLLC solver (\ref{['eqn: chap 5 approx. soln. HLLC']}).
  • Figure 5: Examples of meshes used for the numerical tests: (a) square mesh $h=1/4$ with $16$ fourth-order curved quadrangles with representation of the nodes; (b) example of unstructured mesh in the range $2.5\leqslant x\leqslant 5$ with $1580$ elements and initial positions of the shock and bubble (see \ref{['ssec: Shock in air interacts with a helium bubble']}).
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 6 more