Positivity-preserving high-order discontinuous Galerkin schemes for Ten-Moment Gaussian closure equations

Meena, Asha Kumari; Kumar, Harish; Chandrashekar, Praveen

doi:10.1016/j.jcp.2017.03.024

Cited by 14 publications

(24 citation statements)

References 16 publications

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“…Note that, use of such a limiter at the postprocessing does not guaranty the entropy stability. We also use the bound preserving limiter presented in [26], wherever needed. Bound preserving limiter does not increase the entropy (see [8]).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Following [2,31,26], we consider the following two-dimensional ten-moment Gaussian closure model with source terms:…”

Section: Ten-moment Gaussian Closure Equationsmentioning

confidence: 99%

“…Due to this, the simulations of such flows using the scalar description of the pressure (used most commonly in Euler equations of compressible fluid flows) is not accurate and instead a tonsorial pressure (and hence temperature) is needed. One such model is ten-moment Gaussian closure equations (see [1,2,26,31,24,25,21,22]). It is one of the simplest fluid model which consider pressure as tensor.…”

Section: Introductionmentioning

confidence: 99%

“…In [1,2] Berthon has developed first-order entropy stable and first-order positivity preserving schemes. The positivity preserving schemes are then extended to higher-order finite volume, DG, and WENO schemes by Meena et al in [26,25,27]. Furthermore, in [24], authors have proposed a well-balanced scheme for a potential type source terms.…”

Section: Introductionmentioning

confidence: 99%

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Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

Biswas,

Kumar,

Yadav

2020

Preprint

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In this article, we propose high order discontinuous Galerkin entropy stable schemes for ten-moment Gaussian closure equations, which is based on the suitable quadrature rules (see [8]). The key components of the proposed method are the use of an entropy conservative numerical flux [31] in each cell and a suitable entropy stable numerical flux at the cell edges. This is then used in the entropy stable DG framework of [8] to obtain entropy stability of the semi-discrete scheme. We also extend these schemes to a source term that models plasma laser interaction. For the time discretization, we use strong stability preserving schemes. The proposed schemes are then tested on several test cases to demonstrate stability, accuracy, and robustness.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Following [2,31,26], we consider the following two-dimensional ten-moment Gaussian closure model with source terms:…”

Section: Ten-moment Gaussian Closure Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

Biswas,

Kumar,

Yadav

2020

Preprint

Self Cite

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“…In essence, it is to explore whether or not the range of the high-dimensional function E h is always contained in G: E h (G s+k+1 ) ⊆ G. For some scalar PDEs with linear constraints, for instance, the scalar conservation laws with the constraints linearly defined by maximum principle, a general approach for bound-preserving analysis and design is to exploit certain monotonicity in schemes; see, e.g., [67,14,31]. Yet, for PDE systems especially with nonlinear constraints, there is no unified tool like monotonicity, so that direct and complicated algebraic verification usually has to be performed for each constraint case-by-case for different schemes and different PDEs; see, e.g., [68,38,56,41,66,35,53]. Therefore, the design and analysis of bound-preserving schemes involving nonlinear constraints are highly nontrivial, even for first-order schemes; cf.…”

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confidence: 99%

Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes

Wu¹,

Shu²

2021

Preprint

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Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and is actively studied in recent years. This is however still a challenging task for many systems especially those involving nonlinear constraints.Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transfer all nonlinear constraints into linear ones, through properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions, and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations, and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, by diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.

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A well-balanced scheme for Ten-Moment Gaussian closure equations with source term

Meena

Kumar

2017

Z. Angew. Math. Phys.

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Positivity-preserving high-order discontinuous Galerkin schemes for Ten-Moment Gaussian closure equations

Cited by 14 publications

References 16 publications

Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes

A well-balanced scheme for Ten-Moment Gaussian closure equations with source term

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