[Paper Review] 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+1 more|arXiv (Cornell University)|Nov 29, 2022

Computational Fluid Dynamics and Aerodynamics99 references4 citations

TL;DR

This paper presents a high-order, fully conservative discontinuous Galerkin method for the multicomponent, chemically reacting compressible Euler equations that ensures positivity of density, pressure, and species concentrations, while bounding entropy to suppress nonlinear instabilities. The method combines a linear-scaling limiter with entropy-stable time integration via diagonal-norm summation-by-parts operators, achieving optimal convergence and robustness in one-dimensional test cases including detonation waves and shock tubes.

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.

Motivation & Objective

• Address nonlinear instabilities in high-order simulations of multicomponent, reacting compressible flows with complex thermodynamics.
• Overcome spurious pressure oscillations and negative species concentrations common in conservative schemes.
• Develop a framework that preserves mass, total energy, and atomic element conservation while enforcing physical bounds.
• Ensure discrete satisfaction of the minimum entropy principle to stabilize stiff, reactive flows.
• Extend a prior conservative DG scheme to include positivity and entropy bounding for robust, high-order simulation of detonations and shock waves.

Proposed method

• Prove the minimum entropy principle for the multicomponent, chemically reacting Euler equations, extending prior results from nonreacting flows.
• Implement a linear-scaling limiter that enforces nonnegative species concentrations, positive density, positive pressure, and bounded entropy.
• Ensure the limiter preserves pressure equilibrium between elements by leveraging the consistent flux formulation from Johnson and Kercher (2020).
• Use Strang splitting to decouple convection and stiff chemical source terms, enabling efficient time integration.
• Develop an entropy-stable DG method for ODEs using diagonal-norm summation-by-parts operators and an entropy-conservative two-point numerical flux.
• Apply artificial viscosity to damp small-scale oscillations not fully suppressed by the linear limiter.

Experimental results

Research questions

• RQ1 Can the minimum entropy principle be extended to the multicomponent, chemically reacting compressible Euler equations?
• RQ2 How effective is a linear-scaling limiter in maintaining positivity and bounded entropy in multicomponent reacting flows compared to monocomponent cases?
• RQ3 Does enforcing an entropy bound significantly reduce large-scale nonlinear instabilities in high-order simulations of complex reacting flows?
• RQ4 Can the proposed method achieve optimal high-order convergence while preserving conservation and physical bounds in one-dimensional test cases?
• RQ5 How does the performance of local versus global entropy bounds compare in regions with strong entropy gradients?

Key findings

• The minimum entropy principle holds for the chemically reacting, multicomponent compressible Euler equations, justifying its use as a discrete stability criterion.
• The positivity-preserving limiter prevents solver crashes in shock-tube and thermal-bubble flows, but large-scale oscillations persist without additional stabilization.
• The entropy limiter reduces instabilities more effectively than in monocomponent, calorically perfect flows, demonstrating greater relative benefit in thermally perfect, multicomponent systems.
• Local entropy bounds outperform global bounds in domains with strong spatial entropy variations, such as in detonation waves.
• The method achieves optimal high-order convergence in smooth flows, as demonstrated in the thermal-bubble advection test case.
• Discrete conservation of mass, total energy, and atomic elements is confirmed across all test cases, including the hydrogen-oxygen detonation with detailed chemistry.