Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III

Harten, A.; Engquist, Björn; Osher, Stanley; Chakravarthy, Sukumar

doi:10.1006/jcph.1996.5632

Cited by 324 publications

(238 citation statements)

References 29 publications

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“…Cockburn & Shu (1989), the essentially non-oscillatory (ENO) finite volume and finite difference schemes, e.g. Harten et al (1987) and Shu & Osher (1988), and the weighted ENO (WENO) finite volume and finite difference schemes, e.g. Liu et al (1994) and Jiang & Shu (1996).…”

Section: Introductionmentioning

confidence: 99%

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

2011

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), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov-Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving highorder schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

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Section: Introductionmentioning

confidence: 99%

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

2011

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“…The ENO scheme was developed by Harten et al (1987). The WENO scheme was developed by Liu et al (1994).…”

Section: Weno Schemementioning

confidence: 99%

Third Order Reconstruction of the KP Scheme for Model of River Tinnelva

Dissanayake¹,

Sharma²,

Lie³

2017

MIC

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The Saint-Venant equation/Shallow Water Equation is used to simulate flow of river, flow of liquid in an open channel, tsunami etc. The Kurganov-Petrova (KP) scheme which was developed based on the local speed of discontinuity propagation, can be used to solve hyperbolic type partial differential equations (PDEs), hence can be used to solve the Saint-Venant equation. The KP scheme is semi discrete: PDEs are discretized in the spatial domain, resulting in a set of Ordinary Differential Equations (ODEs). In this study, the common 2 nd order KP scheme is extended into 3 rd order scheme while following the Weighted Essentially Non-Oscillatory (WENO) and Central WENO (CWENO) reconstruction steps. Both the 2 nd order and 3 rd order schemes have been used in simulation in order to check the suitability of the KP schemes to solve hyperbolic type PDEs. The simulation results indicated that the 3 rd order KP scheme shows some better stability compared to the 2 nd order scheme. Computational time for the 3 rd order KP scheme for variable step-length ode solvers in MATLAB is less compared to the computational time of the 2 nd order KP scheme. In addition, it was confirmed that the order of the time integrators essentially should be lower compared to the order of the spatial discretization. However, for computation of abrupt step changes, the 2 nd order KP scheme shows a more accurate solution.

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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

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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Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III

Cited by 324 publications

References 29 publications

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

Third Order Reconstruction of the KP Scheme for Model of River Tinnelva

An artificial neural network as a troubled-cell indicator

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