TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems

Cockburn, Bernardo; Lin, San‐Yih; Shu, Chi‐Wang

doi:10.1016/0021-9991(89)90183-6

Cited by 1,312 publications

(955 citation statements)

References 13 publications

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“…Also, the 800-cell solution is nearly indistinguishable from the exact solution. The results here compare favorably with the TVD results of (Huynh, 1995;Cockburn et al, 1989). …”

Section: Shu-osher Problemsupporting

confidence: 82%

“…A comparison of DG(k) with the multi-stage version (Cockburn et al, 1989;Atkins and Shu, 1996) shows two main advantages in keeping the method compact. The first advantage is that the Courant restriction for the multi-stage version is inversely proportional to IC, at least for k = 1,2.…”

Section: Resultsmentioning

confidence: 99%

“…Note that the semi-discrete version of DG, with (k+ 1)-multi-stage, is O(h"') accurate (Cockburn and Shu, 1989), and therefore does not have the superconvergence property.…”

Section: Accuracymentioning

confidence: 99%

See 2 more Smart Citations

Space-Time Methods for Hyperbolic Conservation Laws

Lowrie

Roe

Leer

1998

ICASE/LaRC Interdisciplinary Series in Science and Engineering

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“…Also, the 800-cell solution is nearly indistinguishable from the exact solution. The results here compare favorably with the TVD results of (Huynh, 1995;Cockburn et al, 1989). …”

Section: Shu-osher Problemsupporting

confidence: 82%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Space-Time Methods for Hyperbolic Conservation Laws

Lowrie

Roe

Leer

1998

ICASE/LaRC Interdisciplinary Series in Science and Engineering

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“…The mass matrix can be made diagonal by choosing the basis functions as Legendre polynomials over each cell [1]. The coefficients of U h (τ, ·) can then be grouped in a vector U (τ ).…”

Section: The Algorithmmentioning

confidence: 99%

Simulation of Gravity Flow of Granular Materials in Silos

Gremaud

Matthews

2000

Lecture Notes in Computational Science and Engineering

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Abstract. The problem of determining the steady state flow of granular materials in silos under the action of gravity is considered. In the case of a Mohr-Coulomb material, the stress equations correspond to a system of hyperbolic conservation laws with source terms and nonlinear boundary conditions. A higher order Discontinuous Galerkin method is proposed and implemented for the numerical resolution of those equations. The efficiency of the approach is illustrated by the computation of the stress fields induced in silos with sharp changes of the wall angle.

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“…In the second step, the solution in these cells are corrected to avoid spurious oscillations. Several correction techniques are available for use in numerical schemes, such as slope limiting [2,3,4], adaptively choosing the stencil for reconstruction [5,6,7], or adding artificial diffusion [8,9,10]. For most numerical methods, the two stages are usually combined into a single step.…”

Section: Introductionmentioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

128

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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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TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems

Cited by 1,312 publications

References 13 publications

Space-Time Methods for Hyperbolic Conservation Laws

Space-Time Methods for Hyperbolic Conservation Laws

Simulation of Gravity Flow of Granular Materials in Silos

An artificial neural network as a troubled-cell indicator

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