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

Computational Fluid Dynamics and Aerodynamics4 citations

TL;DR

This paper presents a high-order, multidimensional discontinuous Galerkin method for chemically reacting compressible flows that enforces positivity of density, pressure, and species concentrations, along with a lower bound on entropy, using a linear-scaling limiter. The method maintains discrete conservation of mass, total energy, and atomic elements, preserves pressure equilibrium across curved elements, and enables robust, accurate simulation of complex detonation waves in 2D and 3D with high-order polynomials on coarse meshes—demonstrating that entropy bounding is essential for stability beyond positivity alone.

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.

Motivation & Objective

• To extend a one-dimensional positivity-preserving and entropy-bounded DG scheme to multiple spatial dimensions for chemically reacting flows.
• To enable robust, high-order simulations of complex reacting flows on relatively coarse meshes without requiring excessive resolution.
• To maintain pressure equilibrium between elements, especially across material interfaces, in multidimensional curved-mesh settings.
• To ensure discrete conservation of mass, total energy, and atomic species in the presence of detailed chemistry and thermodynamic models.
• To demonstrate the critical role of entropy bounding in stabilizing nonlinear solvers during reaction steps, particularly in coarse-mesh detonation simulations.

Proposed method

• The method extends a fully conservative DG formulation to multidimensional, curved elements using arbitrary-shaped, high-order geometric approximations.
• A linear-scaling limiter enforces nonnegative species concentrations, positive density, positive pressure, and a lower bound on specific thermodynamic entropy.
• The scheme is compatible with general quadrature rules (positive weights), arbitrary numerical flux functions, and mixtures of thermally perfect gases.
• Pressure equilibrium is preserved through consistent evaluation of thermodynamic states and nodal basis choices, extending prior work in one dimension.
• Artificial viscosity is applied to suppress small-scale instabilities not eliminated by the entropy and positivity constraints.
• The reaction step uses an operator-splitting approach with a diagonal-norm summation-by-parts DG formulation to treat stiff chemical source terms.

Experimental results

Research questions

• RQ1 Can a multidimensional DG method maintain positivity of density, pressure, and species concentrations while enforcing a lower bound on entropy in chemically reacting flows?
• RQ2 How can pressure equilibrium be preserved across curved, multidimensional elements with arbitrary geometric approximation orders?
• RQ3 What is the impact of entropy bounding versus positivity alone on the stability and convergence of nonlinear solvers in high-order reacting flow simulations?
• RQ4 Can robust and accurate detonation wave simulations be achieved using high-order polynomials on relatively coarse meshes with this formulation?
• RQ5 How does the proposed method compare to existing positivity-preserving or entropy-bounded DG schemes in terms of flexibility and robustness?

Key findings

• The proposed method successfully maintains pressure and velocity equilibrium across curved elements in a 2D thermal bubble advection test case across multiple polynomial orders.
• In 2D and 3D detonation wave simulations, the method achieved stable and accurate solutions using polynomial orders up to p=2 on coarse meshes, whereas prior methods required linear polynomials and very fine meshes.
• The enforcement of an entropy bound was found to be essential for solver convergence; without it, nonlinear solvers stalled or slowed significantly due to large temperature undershoots.
• The method demonstrated optimal high-order convergence in smooth flows, such as thermal-bubble advection, confirming its accuracy.
• Mass, total energy, and atomic species were conserved to machine precision in all test cases, confirming the discrete conservation properties of the scheme.
• Three-dimensional detonation simulations revealed symmetric, in-phase rectangular mode structures with accurate prediction of triple-point line collisions and slapping waves, confirming physical fidelity.