On maximum-principle-satisfying high order schemes for scalar conservation laws

Zhang, Xiangxiong; Shu, Chi‐Wang

doi:10.1016/j.jcp.2009.12.030

Cited by 488 publications

(545 citation statements)

References 28 publications

(47 reference statements)

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“…An arbitrary order of accuracy in space is obtained through the use of the Legendre polynomials hierarchical basis, and low storage strong-stability preserving Runge-Kutta methods (SSP-RK in the following) are used for the time discretization. The whole model is shown to exactly preserve the motionless steady states, thanks to the pre-balanced reformulation of the surface gradient term and suitable interface fluxes [20], and a robust treatment is implemented for the moving shoreline, based on the enforcement of an element-wise water height positivity preservation property, borrowing the recent accuracy-preserving method introduced for the dGM in [79,81]. We lastly introduce an efficient way of handling broken waves, relying on the GN-NSW switching method.…”

Section: Introductionmentioning

confidence: 99%

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

Duran

Marche

2015

Commun. Comput. Phys.

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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...

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Section: Introductionmentioning

confidence: 99%

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

Duran

Marche

2015

Commun. Comput. Phys.

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“…(178), a numerical method should also preserve positivity of the distribution function, which is often en-sured with the use of limiters and a suitable condition on the time step [44]. (For the case of fermions, the distribution function should also remain bounded by 1.)…”

Section: Resultsmentioning

confidence: 99%

Conservative3+1general relativistic Boltzmann equation

2013

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We present a new derivation of the conservative form of the general relativistic Boltzmann equation and specialize it to the 3+1 metric. The resulting transport equation is intended for use in simulations involving numerical relativity, particularly in the absence of spherical symmetry. The independent variables are lab frame coordinate basis spacetime position components and comoving frame curvilinear momentum space coordinates. With an eye towards astrophysical applicationssuch as core-collapse supernovae and compact object mergers-in which the fluid includes nuclei and/or nuclear matter at finite temperature, and in which the transported particles are neutrinos, we examine the relationship between lepton number and four-momentum exchange between neutrinos and the fluid.

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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

117

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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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On maximum-principle-satisfying high order schemes for scalar conservation laws

Cited by 488 publications

References 28 publications

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

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

Conservative3+1general relativistic Boltzmann equation

An artificial neural network as a troubled-cell indicator

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