High order finite difference WENO schemes with the exact conservation property for the shallow water equations

By Yulong Xing, Chi-Wang Shu. 2005

375 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/e455a10d05f6af07b55ed48e1a2728f3934b28da

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In Computational Mechanics

Citations in the corpus

Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows (2006, methodology): “To check the numerical order of accuracy we use the same experiment as Xing and Shu [29].”
Source term treatment of SWEs using surface gradient upwind method (2012, methodology): “Caleffi et al. (2006, 2007), Xing and Shu (2005), Liang and Borthwick (2009), and Canestrelli et al. (2010) also further improved the scheme by using the weighted essentially non-oscillatory method, the discontinuous Galerkin method, the adaptive-quadtree method, and the primitive centred method,…”
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems (2009, methodology): “High order finite difference and finite volume WENO schemes which are well balanced for the still water solution of the shallow water equations, and a more general class of balance laws, were designed in [117, 168, 171, 172, 173, 174].”
High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms (2006, methodology): “Different from [33,34], a key ingredient of the technique used in [35] is a special decomposition of the source term, allowing a discretization to the source term to be both high order accurate for general solutions and exactly well balanced with the flux gradient for still water.”
Godunov-type methods for free-surface shallow flows: A review (2007, methodology): “These schemes are applicable to the shallow water type equations, see Crnjaric-Zic et al., (2004), Noelle et al. (2006) and Xing and Shu (2005).”
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, positive): “We have the following bottom function and initial conditions b(x) = sin(2)(πx), h(x, 0) = 5 + e, (hu)(x, 0) = sin(cos(2πx)), x ∈ [0, 1] with periodic boundary conditions, see [31].”
A SECOND-ORDER WELL-BALANCED POSITIVITY PRESERVING CENTRAL-UPWIND SCHEME FOR THE SAINT-VENANT SYSTEM ∗ (2007, methodology): “The methods that exactly preserve such solutions are called well-balanced, and we refer the reader to [2, 4, 8, 10, 12, 16, 17, 20, 21, 22, 23, 27, 28], where a variety of highorder well-balanced schemes for the Saint-Venant system can be found.”
High-order balanced CWENO scheme for movable bed shallow water equations (2007, methodology): “This is verified using a movable bed case obtained modifying the accuracy test proposed by Xing and Shu, [37].”