On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

Zhang, Xiangxiong; Shu, Chi‐Wang

doi:10.1016/j.jcp.2010.08.016

Cited by 565 publications

(618 citation statements)

References 30 publications

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“…The viscous approximations for regularized conservation laws, written in general as 31) must also satisfy the SBP condition. Integration by parts yields…”

Section: Variable Coefficient Second Derivative Approximationmentioning

confidence: 99%

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High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Fisher

Carpenter

2013

Journal of Computational Physics

283

265

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“…The viscous approximations for regularized conservation laws, written in general as 31) must also satisfy the SBP condition. Integration by parts yields…”

Section: Variable Coefficient Second Derivative Approximationmentioning

confidence: 99%

“…Another mechanism must be employed to bound ρ and T away from zero to ensure stability. This is not considered in the present work, but interested readers can refer to the work by Zhang and Shu [31]. The full derivation of the entropy variables and symmetrizing matrices are detailed in Appendix B.1.…”

Section: Entropy Analysismentioning

confidence: 99%

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Fisher

Carpenter

2013

Journal of Computational Physics

283

265

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“…The MLP indicator did not face this issue for any of the Euler test cases considered in this paper. However, it might be useful to add a positivity preserving limiter [48,49] for more complex test cases.…”

Section: Left Half Of the Blast-wavementioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

120

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High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

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“…where (ρE) e i is given by (30). The corresponding correction factor α e,ρE is calculated using (29) to enforce the total energy constraints (36).…”

Section: Limiting For the Euler Equationsmentioning

confidence: 99%

“…Positivity-preserving pressure limiters were developed for high-order discontinuous Galerkin discretizations of the Euler equations in a number of recent publications by Zhang and Shu [30,31]. Whereas positivity preservation is a necessary condition for obtaining physically realistic solutions, it does not rule out the presence of undershoots and/or overshoots in the pressure distribution.…”

Section: Introductionmentioning

confidence: 99%

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

Dobrev

Kolev

Kuzmin

et al. 2018

Journal of Computational Physics

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We present a new approach to enforcing local maximum principles in finite element schemes for the compressible Euler equations. In contrast to synchronized limiting techniques for systems of conservation laws, the density, momentum, and total energy are constrained in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum are adjusted to incorporate the irreversible effect of density changes. Then the corresponding antidiffusive corrections are limited to satisfy inequality constraints for the total and kinetic energy. The same element-based limiting strategy is employed in the context of continuous and discontinuous Galerkin methods. The sequential nature of the new limiting procedure makes it possible to achieve crisp resolution of contact discontinuities while using sharp local bounds in the energy constraints. A numerical study is performed for piecewise-linear finite element discretizations of 1D and 2D test problems.

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On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

Cited by 565 publications

References 30 publications

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

An artificial neural network as a troubled-cell indicator

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

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