A technique of treating negative weights in WENO schemes

By Jing Shi, Changqing Hu, Chi-Wang Shu. 2000

428 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/3c418f6f91ffcf2270d9618c4a7245baf1c61a88

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

ADER schemes for three-dimensional non-linear hyperbolic systems (2005, methodology): “1–6 and [22]) it is seen that the rolling of slip surfaces is much more pronounced in the results of ADER4-HLL scheme.”
A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms (2006, neutral): “The complete algorithm can be found in [16,23,25].”
Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows (2006, methodology): “It is well known that the negative weights appearing in (37) may lead to oscillations at discontinuities [ 22 ].”
Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters (2005, methodology): “In this paper we use the usual WENO reconstructions based on cell averages of neighboring cells, such as in [21, 16], to reconstruct the values of the solutions at certain Gaussian quadrature points in the troubled cells, and then rebuild the solution”
Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments (2011, background): “found in the appendix of Zhang & Shu (2010b), where the negative linear weights should be dealt with by the method in Shi et al. (2002). Extensive tests for scalar conservation laws were done to test the accuracy for the new limiter mentioned above. The results are similar to those in Zhang & Shu (2”
Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments (2014, methodology): “[119] designed WENO limiters using the usual WENO reconstruction based on cell averages of neighboring cells as in [52, 40, 78], to reconstruct the values of the solutions at certain Gaussian quadrature points in the target cells, and then rebuild the solution polynomials from the original cell aver”
Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case (2004, methodology): “We have the third-order finite volume WENO schemes in one space dimension in [18], the third and fifth-order finite difference WENO schemes in multi-space dimensions with a general framework for the design of the smoothness indicators and non-linear weights in [16], finite difference WENO schemes of”
Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems (2007, methodology): “For a history of developments concerning high order accurate polynomial recovery for finite volume methods on unstructured triangulations see the references [1,4,17,25,37,30,33,41,43], and for high order ENO and WENO methods on structured grids see for instance [3,28,31].”
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems (2009, methodology): “Finite volume WENO schemes on two-dimensional unstructured meshes were designed in [50, 73, 146, 44, 45].”
HIGH ORDER NUMERICAL METHODS FOR TIME DEPENDENT HAMILTON-JACOBI EQUATIONS (2007, methodology): “We omit the details of this technique and refer the readers to [34].”
Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials (2012, methodology): “Several strategies became very popular due to their intrinsic simplicity such that the MUSCL method [3,6,7,20,26,32] or their efficiency to achieve very high-order accuracy such that the ENO/WENO method [1,21,29,30,36,37,46,18,19,35,49,41], the Discontinuous Galerkin method [9– 12], the ADER method ”
High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms (2006, methodology): “[22], see also [16,27], and Runge–Kutta discontinuous Galerkin (RKDG) finite element methods that were originally developed by Cockburn and Shu [8], see also [7,9].”
Unstructured Large-Eddy Simulations of Supersonic Jets (2017, methodology): “perform reconstructions [30] as inputs to the HLLC approximate Riemann [28] solver to compute the flux across a shock.”
On the construction, comparison, and local characteristic decomposition for high-Order central WENO schemes (2002, methodology): “[25]: We first split the linear weights into two groups”
Finite-volume WENO schemes for three-dimensional conservation laws (2004, methodology): “The procedure to evaluate numerical fluxes (6) in three space dimensions is a straightforward extension of the corresponding two-dimensional one [2, 10, 19, 6] and consists of three main steps.”
On maximum-principle-satisfying high order schemes for scalar conservation laws (2010, methodology): “3.3 Implementation for the fifth order finite volume WENO scheme First, Let us recall the dimension by dimension reconstruction procedure of the finite volume WENO for (3.1) in [22] [24].”
High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD (2001, methodology): “For structured meshes, finite volume WENO schemes of fifth order accuracy Shi et al., 2002 are very robust and allow for arbitrary, non-smooth mesh sizes, hence, they can be used in more general situations than the finite difference WENO schemes.”
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems (2007, methodology): “The initial condition for the system is zero except of the first component (density fluctuation q) which is given according to Jiang and Shu [25] as qðx; 0Þ ¼ 1 6 ½Gðx; z dÞ þ Gðx; zþ dÞ þ 4Gðx; zÞ ; 0:8 6 x 6 0:6; 1; 0:4 6 x 6 0:2; 1 j10 x 0:1ð Þj; 0:0 6 x 6 0:2; 1 6 ½F ðx; a dÞ þ F ðx; aþ dÞ þ 4F ”