A Comparison of Troubled-Cell Indicators for Runge--Kutta Discontinuous Galerkin Methods Using Weighted Essentially Nonoscillatory Limiters

Qiu, Jianxian; Shu, Chi‐Wang

doi:10.1137/04061372x

Cited by 172 publications

(137 citation statements)

References 27 publications

Supporting

Mentioning

134

Contrasting

Unclassified

Order By: Relevance

“…Waves about to break are identified in elements such that I j > 1 (called troubled elements, following [58]) and we locally suppress the dispersive term in such elements (i.e. we locally switch to the NSW equations).…”

Section: Handling Broken Waves and Limiting Strategymentioning

confidence: 99%

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Duran

Marche

2015

Commun. Comput. Phys.

View full text Add to dashboard Cite

Abstract. We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using an expansion basis of arbitrary order. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to handle broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.. . IntroductionDepth-averaged equations are widely used in coastal engineering for the simulation of nonlinear waves propagation and transformations in nearshore areas. The full description of surface water waves in an incompressible, homogeneous, inviscid fluid, is provided by the free surface Euler (or water waves) equations but this problem remains mathematically and numerically challenging. As a consequence, the use of depth averaged equations helps to reduce the three-dimensional problem to a two-dimensional problem, while keeping a good level of accuracy in many configurations. Many Boussinesq-like models are used nowadays and a detailed review can be found in [42] and the recent monograph [41]. Denoting by λ the typical horizontal scale of the flow and h 0 the typical depth, the shallow water regime usually corresponds to the configuration where µ := [52,54,57] for instance, an additional smallness amplitude assumption on the typical wave amplitude a is classically performed: ε := a h0 = O(µ). This assumption often appears as too restrictive for many applications in coastal oceanography. Removing the small amplitude assumption while still keeping all the O(µ) terms, we obtain the so-called Green-Naghdi equations (GN equations in the following) [34], also referred to as Serre equations [62] or fully non-linear Boussinesq equations [76]. A large number of numerical methods have been developed in the past few years for the BT equations. Let us mention for instance some Finite-Difference (FDM in the following) approaches [49,54,65,75] As far as flexibility is concerned, the use of discontinuous-Galerkin methods (dG methods in the following) would appear as a natural choice. Indeed, this class of method provides several appealing features, like compact discretization stencils and hp-adaptivity, flexibility with a natural handling of unstructured meshes, easy parallel computation and local conservation properties in the approximation of conservation laws. A general review of dG methods for convection dominated problems is performed in [14]. Concerning the approximation of more general problems, involving higher-order derivatives, several methods a...

show abstract

Section: Handling Broken Waves and Limiting Strategymentioning

confidence: 99%

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Duran

Marche

2015

Commun. Comput. Phys.

View full text Add to dashboard Cite

show abstract

“…Note that the original troubled-cell indicators are applied using the optimal problem-dependent parameters as found in [28,39]. We stress that the outlier-detected results are computed without problem-dependent parameters, but with a fixed whisker length equal to 3, and with local indication vectors of size 16. It turns out that the new outlier-detection approach detects the troubled regions very accurately and generally better than the original parameter-using methods for the blast-wave and Shu-Osher problem.…”

Section: One-dimensional Testsmentioning

confidence: 99%

“…We omit the details of these test problems and refer the reader to [39] for more information on initial conditions and boundary conditions. We apply the indication technique to density for the modified multiwavelet indicator, density and energy for KXRCF, and the characteristic variables for the minmod-based TVB indicator, as done by Qiu and Shu [28]. The first row of each figure consists of time-history plots of detected troubled cells using the original indicators.…”

Section: One-dimensional Testsmentioning

confidence: 99%

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

Vuik

Ryan

2016

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. In Vuik and Ryan [J. Comput. Phys., 270 (2014), pp. 138-160] we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has an impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in the detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond the others [J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, 1977]. We provide an algorithm that can be applied to various troubled-cell indication variables. With this technique, the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically.

show abstract

“…A thorough numerical comparison of various limiter-based indicator functions has been performed in [12]. We consider the following two commonly used limiters: 1.…”

Section: Discontinuous-galerkin Formulationmentioning

confidence: 99%

“…To ensure cost-efficiency of numerical schemes, it is essential to use troubled-cell indicators that only flag the genuine troubled-cells. In [12], a thorough numerical study was performed to assess the performance of various limiter-based troubled-cell indicators for RKDG schemes. It was observed that the classical minmod limiter flags more cells than necessary, including cells with smooth extrema, which can lead to an unnecessary increase in computational cost.…”

Section: Introductionmentioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

117

View full text Add to dashboard Cite

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.

show abstract

A Comparison of Troubled-Cell Indicators for Runge--Kutta Discontinuous Galerkin Methods Using Weighted Essentially Nonoscillatory Limiters

Cited by 172 publications

References 27 publications

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

An artificial neural network as a troubled-cell indicator

Contact Info

Product

Resources

About