Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method, III: Unstructured Meshes

Zhu, Jun; Qiu, Jianxian

doi:10.1007/s10915-009-9271-7

Cited by 104 publications

(40 citation statements)

References 26 publications

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“…In this paper we would like to obtain the third-order accuracy in time, so we need to approximate those terms on the right hand side in (27) until the third time derivatives and neglect other terms behind the third time derivatives. Theoretically, we can obtain any accuracy order in time and just need to approximate those terms until the according time derivatives.…”

Section: The Lax-wendroff-type Discretization For 1d Shallowmentioning

confidence: 99%

“…In recent years, WENO scheme has been generalized for hyperbolic conservation laws [22,26,27] after the first WENO scheme was originally derived in [28] for the thirdorder finite volume frame based on ENO type schemes [29,30], such as CWENO and hybrid WENO [31] arising from various applications. In 2016, a successful type of WENO [32] was proposed to approximate hyperbolic conservation laws.…”

Section: Introductionmentioning

confidence: 99%

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The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

Xie

Yang

2018

Mathematical Problems in Engineering

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A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

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Section: The Lax-wendroff-type Discretization For 1d Shallowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

Xie

Yang

2018

Mathematical Problems in Engineering

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“…This way keeps the compact property but would damage the accuracy in smooth region if marked as trouble cells. The other is to develop high-order limiting technique such as WENO/Hermite WENO [48] for discontinuous Galerkin method. The accuracy of this way can improve comparing to second order limiter, but it destroys the compactness of the original methods.…”

Section: Shock Capturing Techniquementioning

confidence: 99%

Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids

2014

Springer Theses

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“…In [25], the HWENO methodology was also proposed as a limiter for the RKDG schemes. The HWENO method is extended to twodimensional problems on a structured grid in [26] and on an unstructured grid in [27]. A further improvement of two-dimensional HWENO schemes is presented in [28], where a robust treatment for problems involving strong shock was given.…”

Section: Introductionmentioning

confidence: 99%

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Caleffi

2010

Numerical Methods in Fluids

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Hermite weighted essentially non-oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high-order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193:115). For example, classical fifth-order weighted essentially non-oscillatory (WENO) reconstructions are based on a five-cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three-cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth-order accurate well-balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five-step and fourth-order accurate method. Besides the classical SWE, the non-homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well-balanced treatment of the source term involved in such equations is developed and tested. Several standard one-dimensional test cases are used to verify the high-order accuracy, the C-property and the good resolution properties of the model

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Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method, III: Unstructured Meshes

Cited by 104 publications

References 26 publications

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

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