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

Tao, Zhanjing; Li, Fengyan; Qiu, Jianxian

doi:10.1016/j.jcp.2016.05.005

Cited by 27 publications

(9 citation statements)

References 39 publications

(85 reference statements)

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“…Then, the 1-D MLC scheme can be extended to two-dimensional cases naturally. Similar idea can be found in the central Hermite WENO scheme developed by Tao et al 49 , though their method is based on the finite volume framework. In their methods, the equation of the mixed first-order momentum is introduced when the dimension is higher than one.…”

Section: Introductionmentioning

confidence: 84%

Very High-Order Upwind Multi-Layer Compact (MLC) Schemes with Spectral-Like Resolution II: Two-Dimensional Case

Bai

Zhong

2019

AIAA Scitech 2019 Forum

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In this paper, the very high-order upwind multi-layer compact (MLC) scheme developed by Bai and Zhong 1 is revisited and analyzed with focus on the two-dimensional case. The MLC scheme is designed to solve smooth multi-scale flow problems with complex physics, such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics, etc. In its multi-layer framework, the auxiliary equations for the first derivatives are introduced. Accordingly, the first derivatives are evolved simultaneously with the function values. The MLC scheme derived on structured grids has an explicit finite difference formulation, which includes both function values and first derivatives. Benefiting from the multi-layer framework, the scheme achieves very high-order accuracy and spectral-like resolution within a compact stencil. The upwind MLC scheme is derived on a centered stencil, with an adjustable parameter to introduce small dissipation for stability. A main problem of the original MLC scheme 1 is the minor numerical instability in the twodimensional case, which is mainly triggered by the inconsistency between the 1-D and 2-D MLC formulations. Also, the approximation of the cross derivative in the original scheme introduces uncertainty, and it is relatively expensive in very high-order cases. In this paper, a directional discretization technique is designed to extend the 1-D MLC scheme to twodimensional cases. By introducing the auxiliary equation for the cross derivative, the spatial discretization can be fulfilled along each dimension independently. Therefore, the 1-D MLC scheme can be applied to all spatial derivatives, and the 2-D MLC scheme is not required any more. This directional discretization technique avoids any inconsistency between the 1-D and 2-D MLC schemes, and it also overcomes the uncertainty arising from the approximation of cross derivatives. The Fourier analysis demonstrates that all modes of the new MLC scheme are stable in two-dimensional cases, and the new scheme has better spectral resolution and smaller anisotropic error for a large portion of wavenumbers in [0, 2π]. The analysis through matrix method indicates that stable boundary closure schemes are also easier to be obtained for the new MLC scheme with the directional discretization. The numerical results validate that the new MLC scheme has smaller error and better computational efficiency than the original MLC scheme due to the better spectral resolution. The long-time simulation results verify that the original MLC scheme could be unstable in some cases; while the new MLC scheme with the directional discretization is always stable for both periodic and non-periodic boundary conditions. High-Order Methods There are plenty of high-order numerical methods developed in the past few decades, such as spectral methods 2-4 , compact finite difference schemes 5-12 , discontinuous-Galerkin (DG) methods 13-16 , and nonlinear schemes like TVD 17,18 , ENO 19 , WENO 20,21. Many reviews are available for study on high-order methods. Ekaterinaris 2...

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Section: Introductionmentioning

confidence: 84%

Very High-Order Upwind Multi-Layer Compact (MLC) Schemes with Spectral-Like Resolution II: Two-Dimensional Case

Bai

Zhong

2019

AIAA Scitech 2019 Forum

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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%

“…To make the stencil more compact, Qiu and Shu [28,29] developed the WENO methodology, which were first taken as limiters for Runge-Kutta discontinuous Galerkin methods, termed as Hermite WENO (HWENO) schemes. After this, many HWENO schemes were developed for solving hyperbolic conservations laws [38,7,16,34,24,42,35,8,25]. The HWENO schemes can achieve higher order accuracy than standard WENO schemes on the same stencils.…”

Section: Introductionmentioning

confidence: 99%

“…Since the solutions of nonlinear hyperbolic conservation laws often contain discontinuities, the derivative equations for the HWENO schemes need to deal with the derivatives or the first order moments, which would be relatively large nearby discontinuities. Therefore, the HWENO schemes listed by [28,29,38,34,24,42,35,8,25] all used different stencils to discretize the space for the original equations and the derivative equations, respectively. The variables of the derivative equations for the hybrid HWENO scheme are the first order moments, which also can be seen in discontinuous Galerkin (DG) methods [1,2,3,4] and other HWENO schemes [40,26,35].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the HWENO schemes listed by [28,29,38,34,24,42,35,8,25] all used different stencils to discretize the space for the original equations and the derivative equations, respectively. The variables of the derivative equations for the hybrid HWENO scheme are the first order moments, which also can be seen in discontinuous Galerkin (DG) methods [1,2,3,4] and other HWENO schemes [40,26,35]. In one sense, these HWENO schemes can be seen as an extension by DG methods, and Dumbser et al [9] gave a general and unified framework to define the numerical scheme extended by DG method, termed as P N P M method, in which P N represents a piecewise polynomial of degree N used as test functions in DG method and P M is a polynomial of degree M reconstructed by the test functions of degree N for computing the numerical fluxes, and M ≥ N .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

Tao

Qiu

2017

Adv Comput Math

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

Cited by 27 publications

References 39 publications

Very High-Order Upwind Multi-Layer Compact (MLC) Schemes with Spectral-Like Resolution II: Two-Dimensional Case

Very High-Order Upwind Multi-Layer Compact (MLC) Schemes with Spectral-Like Resolution II: Two-Dimensional Case

A hybrid Hermite WENO scheme for hyperbolic conservation laws

Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

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