High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws

Tao, Zhanjing; Li, Fengyan; Qiu, Jianxian

doi:10.1016/j.jcp.2014.10.027

Cited by 31 publications

(24 citation statements)

References 36 publications

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“…As the solutions of nonlinear hyperbolic conservation laws often contain discontinuities, its derivatives or first order moments would be relatively large nearby discontinuities. Hence, the HWENO schemes presented in [23,24,31,28,21,33,29,7] used different stencils for discretization in the space for the original and derivative equations, respectively. In one sense, these HWENO schemes can be seen as an extension by DG methods, and Dumbser et al [8] gave a general and unified framework to define the numerical scheme extended by DG method, termed as P N P M method.…”

Section: Introductionmentioning

confidence: 99%

A hybrid Hermite WENO scheme for hyperbolic conservation laws

Zhao

Chen

Qiu

2020

Journal of Computational Physics

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In this paper, a fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme with artificial linear weights is proposed for one and two dimensional hyperbolic conservation laws, where the zeroth-order and the first-order moments are used in the spatial reconstruction. We construct the HWENO methodology using a nonlinear convex combination of a high degree polynomial with several low degree polynomials, and the associated linear weights can be any artificial positive numbers with only requirement that their summation equals one.The one advantage of the HWENO scheme is its simplicity and easy extension to multidimension in engineering applications for we can use any artificial linear weights which are independent on geometry of mesh. The another advantage is its higher order numerical accuracy using less candidate stencils for two dimensional problems. In addition, the HWENO scheme still keeps the compactness as only immediate neighbor information is needed in the reconstruction and has high efficiency for directly using linear approximation in the smooth regions. In order to avoid nonphysical oscillations nearby strong shocks or contact discontinuities, we adopt the thought of limiter for discontinuous Galerkin method to control the spurious oscillations. Some benchmark numerical tests are performed to demonstrate the capability of the proposed scheme.

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Section: Introductionmentioning

confidence: 99%

A hybrid Hermite WENO scheme for hyperbolic conservation laws

Zhao

Chen

Qiu

2020

Journal of Computational Physics

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“…In [34], we proposed high-order central HWENO (C-HWENO) schemes for solving the hyperbolic conservation laws in one and two space dimensions. The methods use HWENO reconstructions as spatial discretizations, and Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods as time discretizations, in a central finite volume formulation on staggered meshes.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The work in this paper is a continuation of the development in [34], with the same focus on the design of C-HWENO methods on structured meshes for hyperbolic conservation laws. In [34], the HWENO reconstructions are based on the solution and its derivative(s), while in the present work, we propose new HWENO reconstructions which are based on the zeroth-order and the first-order moments of the solution. Using moments of the solution itself is not new in HWENO reconstruction, with examples including the limiting procedure for DG methods in [28,29], the hybrid RKDG-HWENO reconstruction in [2], and P N P M methods in [8,24].…”

Section: Introductionmentioning

confidence: 99%

“…This will improve the computational efficiency, as well as the ease of implementation in higher dimensions. Meanwhile, the proposed methods have the similar advantages as the C-HWENO methods in [34]. Like in [34], we apply in time the Lax-Wendroff type discretizations [11,18,27] and the natural continuous extension of Runge-Kutta methods [4,19,20,26,37].…”

Section: Introductionmentioning

confidence: 99%

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High-order central Hermite WENO schemes: Dimension-by-dimension moment-based reconstructions

Tao

Qiu

2016

Journal of Computational Physics

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In this paper, a class of high-order central finite volume schemes is proposed for solving one-and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. Different from the central Hermite WENO methods we developed previously in [J. Comput. Phys. 281:148-176, 2015], the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of oneand two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions.

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Implementation of weighted essentially non‐oscillatory scheme for unsteady and steady compressible flow in pressure‐based algorithm

Balaj

Djavareshkian

2021

Numerical Methods in Fluids

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A pressure-based semi-implicit procedure has been developed for the computation of compressible flows on a system with collocated finite volume formulations. The method includes the implementation of a high-order accurate weighted essentially non-oscillatory scheme within the pressure-based algorithm which employs an approximate Riemann solver for the computation of the cell face inviscid fluxes. The developed scheme is applied to the computation of unsteady compressible flow through a shock tube which experiences various Mach numbers, and the computations are compared with an analytical solution. The computations indicate that the developed procedure leads to sharp discontinuity representation without creation of spurious oscillations. Then, the developed numerical method is utilized to investigate unsteady two-dimensional flow for a lax configuration and the results are compared with the results of the similar scheme in the density-based algorithm. At the end, the investigation of steady compressible flows in bump and wedge channel is considered. The simulations show a considerable improvement over the existing pressure-based method. The investigations show that the developed method is suitable for studying compressible steady and unsteady flows.

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High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws

Cited by 31 publications

References 36 publications

A hybrid Hermite WENO scheme for hyperbolic conservation laws

A hybrid Hermite WENO scheme for hyperbolic conservation laws

High-order central Hermite WENO schemes: Dimension-by-dimension moment-based reconstructions

Implementation of weighted essentially non‐oscillatory scheme for unsteady and steady compressible flow in pressure‐based algorithm

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