High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

Gao, Zhen; Hu, Guanghui

doi:10.4208/cicp.oa-2016-0200

Cited by 14 publications

(3 citation statements)

References 38 publications

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“…Caleffi [3] developed a well-balanced fourth-order finite volume Hermite WENO scheme for the one-dimensional SWEs on the basis of [19]. For more related well-balanced high-order methods, e.g., finite difference schemes [6,7,8,14,16,25], finite volume schemes [5,11,18,28], and DG methods [12,30,31,33,34,35].…”

Section: Introductionmentioning

confidence: 99%

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

Zhao¹,

Zhang²

2022

Preprint

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In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balance property. Instead of using a five-point stencil in the same fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction.Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

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Section: Introductionmentioning

confidence: 99%

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

Zhao¹,

Zhang²

2022

Preprint

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“…Many shock capturing schemes with explicit time discretizations have been developed to solve the SWEs with source term (1.1), including high order finite difference [20,36,58,59,62], finite volume [1,10,34,38,43,44,64], residual distribution methods [46,47] and discontinuous Galerkin schemes [60,61,65,67], and many references therein. When solving the SWEs with source term numerically, it is important to preserve the exact conservation property (C-property) [35], namely, the nonzero flux gradient should be exactly balanced by the source term in the case of a stationary water.…”

Section: Introductionmentioning

confidence: 99%

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

Huang,

Xing,

Xiong

2021

Preprint

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In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one-and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

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“…Rogers et al, 2003 andBorthwick, 2009). More recently, Xing and Shu (2005)'s ideas have been further extended to more advanced approaches such as hybrid WENO (Zhu et al, 2017) and weighted compacted nonlinear (WCN) schemes (Gao and Hu, 2017). Li et al (2015) extended Xing and Shu (2005)'s well-balanced strategy to the 'pre-balanced' shallow water equations proposed by Rogers et al (2003), and introduced a robust method that simultaneously combined both well-balanced strategies.…”

Section: Introductionmentioning

confidence: 99%

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

2020

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A numerical wave tank equipped with a piston type wave-maker is presented for long-duration simulations of long waves in shallow water. Both wave maker and tank are modelled using the nonlinear shallow water equations, with motions of the numerical piston paddle accomplished via a linear mapping technique. Three approaches are used to increase computational efficiency and accuracy. First, the model satisfies the exact conservation property (C-property), a stepping stone towards properly balancing each term in the governing equation. Second, a high-order WENO method is used to reduce accumulation of truncation error. Third, a cut-off algorithm is implemented to handle contaminated digits arising from round-off error. If not treated, such errors could prevent a numerical scheme from satisfying the exact C-property in long-duration simulations. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and shock-capturing ability of the present scheme. Correct implementation of the wave paddle generator is verified by comparing numerical predictions against analytical solutions of sinusoidal, solitary, and cnoidal waves. In all cases, the model gives satisfactory results for small-amplitude, low frequency waves. Error analysis is used to investigate model limitations and derive a user criterion for long wave generation by the model.

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High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

Cited by 14 publications

References 38 publications

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

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