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

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

doi:10.48550/arxiv.2211.16254

Cited by 3 publications

(19 citation statements)

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“…S AV is a shock sensor based on solution variations inside a given element [43], C AV is a user-defined parameter, h is a length scale associated with the element, and R (y, ∇y) is the strong form of the residual (2.1). In our previous work, this artificial viscosity formulation effectively mitigated small-scale nonlinear instabilities in various multicomponent-flow problems [2,10]. However, other types of artificial viscosity or limiters can be employed instead.…”

Section: Discontinuous Galerkin Discretizationmentioning

confidence: 99%

“…where G (y) is the homogeneity tensor [33], obtained by differentiating the viscous flux with respect to the gradient, i.e., G(y) = ∂F v /∂∇y. Additional information on the thermodynamic relations, transport properties, and chemical reaction rates can be found in [2] and [10].…”

Section: Governing Equationsmentioning

confidence: 99%

“…Equations ( 3.3) and (3.5) are advanced in time using a strong-stability-preserving Runge-Kutta method (SSPRK) [37,38], whereas Equation (3.4) is solved using a fully implicit, temporal DG discretization. Since the reaction step is identical between the inviscid and viscous cases, we refer the reader to [10] and [11] for more details on the DG discretization in time for Equation (3.4). Here, we focus on the transport step.…”

Section: Discontinuous Galerkin Discretizationmentioning

confidence: 99%

“…Since ρu * (y) is a concave function of the state [10], G is a convex set. If all species concentrations are strictly positive, then for a given σ, χ σ is concave [12,10,9] and G σ is also a convex set. However, if any of the species concentrations is zero, then χ σ is no longer concave [9,46].…”

Section: Transport Step: Positivity-preserving Entropy-bounded Discon...mentioning

confidence: 99%

“…To further increase robustness, we made key advancements to this fully conservative DG method, focusing on the inviscid case, to ensure satisfaction of the positivity property (i.e., nonnegative species concentrations, positive density, and positive pressure) and an entropy bound based on the minimum entropy principle for the multicomponent Euler equations [9,10], which states that the spatial minimum of specific thermodynamic entropy of entropy solutions is a nondecreasing function of time. The main ingredients of this DG formulation [10,11] are (a) an invariant-region-preserving inviscid flux function [12], (b) a simple linear-scaling limiter [13], (c) satisfaction of a time-step-size constraint for the transport step with a strong-stability-preserving explicit time integrator, (d) incorporation of the pressure-equilibrium-preserving techniques introduced by Johnson and Kercher [2], and (e) an entropy-stable DG discretization in time based on diagonal-norm summation-by-parts operators for the reaction step. It was found that the formulation was capable of robustly and accurately computing complex inviscid, reacting flows using high-order polynomials and relatively coarse meshes.…”

Section: Introductionmentioning

confidence: 99%

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Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Ching

Johnson

Burrows

et al. 2023

AIAA SCITECH 2023 Forum

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This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also discuss in detail how to account for boundary conditions and incorporate previously developed pressure-equilibrium-preserving techniques into the positivity-preserving framework. Comparisons between the Lax-Friedrichs-type viscous flux function and more conventional flux functions are provided, the results of which motivate an adaptive solution procedure that employs the former only when the element-local solution average has negative species concentrations, nonpositive density, or nonpositive pressure. The resulting formulation is compatible with curved, multidimensional elements of arbitrary shape and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction. We find that just as in the inviscid, multicomponent case, the robustness benefits of the enforcement of an entropy bound are much more pronounced than in the monocomponent, calorically perfect setting. Where appropriate, we demonstrate that mass, total energy, and atomic elements are discretely conserved.

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Section: Discontinuous Galerkin Discretizationmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Section: Discontinuous Galerkin Discretizationmentioning

confidence: 99%

Section: Transport Step: Positivity-preserving Entropy-bounded Discon...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Ching

Johnson

Burrows

et al. 2023

AIAA SCITECH 2023 Forum

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Kinetic-Energy- and Pressure-Equilibrium-Preserving Schemes for Real-Gas Turbulence in the Transcritical Regime

Bernades¹,

Jofre

Capuano

2023

Preprint

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The Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Robust Simulations of High-Speed Viscous Flows

Ching

Kercher

Corrigan

2023

AIAA SCITECH 2023 Forum

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The moving discontinuous Galerkin method with interface condition enforcement (MDG-ICE) is a highorder, r-adaptive method that treats the grid as a variable and weakly enforces the conservation law, constitutive law, and corresponding interface conditions in order to implicitly fit high-gradient flow features. In this paper, we introduce nonlinear solver strategies to more robustly and efficiently compute high-speed viscous flows. Specifically, we incorporate an anisotropic grid regularization based on the mesh-implied metric into the nonlinear least-squares solver that inhibits grid motion in directions with small element length scales. Furthermore, we develop an adaptive elementwise regularization strategy that locally scales the regularization terms as needed to maintain grid validity. We apply the proposed MDG-ICE formulation to test cases involving viscous shocks and/or boundary layers, including Mach 17.6 hypersonic viscous flow over a circular cylinder and Mach 5 hypersonic viscous flow over a sphere, which are very challenging test cases for conventional numerical schemes on simplicial grids. Even without artificial dissipation, the computed solutions are free from spurious oscillations and yield highly symmetric surface heat-flux profiles.

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

Cited by 3 publications

References 0 publications

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Kinetic-Energy- and Pressure-Equilibrium-Preserving Schemes for Real-Gas Turbulence in the Transcritical Regime

The Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Robust Simulations of High-Speed Viscous Flows

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