A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws

Zhao, Zhenzhen; Qiu, Jianxian

doi:10.1007/s10915-020-01347-1

Cited by 18 publications

(29 citation statements)

References 33 publications

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“…Compared with the standard WENO scheme, its boundary treatment is much simpler and the numerical error is observed to be smaller with the same mesh, as shown in [30]. The HWENO scheme was later extended to solve the hyperbolic conservation laws in the finite difference [21,43] and finite volume [28,29] frameworks, and the same advantages have been observed.…”

Section: Introductionmentioning

confidence: 89%

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state $S_{N}$ transport equation

Ren¹,

Xing²,

Wang³

et al. 2021

Preprint

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In this paper, we propose to combine the fifth order Hermite weighted essentially nonoscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steadystate S N transport equation in the finite volume framework. It is well-known that the S N transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonstrate the asymptotic preserving property of the high order finite volume HWENO method, by showing that its cell-edge and cell-average fluxes possess the thick diffusion limit. Numerical results in both one-and two-dimensions are presented to validate its asymptotic preserving property. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save computational cost. Extensive one-and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM.

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Section: Introductionmentioning

confidence: 89%

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state $S_{N}$ transport equation

Ren¹,

Xing²,

Wang³

et al. 2021

Preprint

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“…Ma and Wu [14] developed a compact HWENO scheme by solving the derivatives using the compact difference method. Recently, Zhao et al [25] developed a genuine fifth-order modified finite difference HWENO (M-HWENO) scheme in one and two dimensions. In [25], the authors modified the derivatives of the solution by a high-order Hermite limiter to control the derivatives near discontinuities and improve the stability of the scheme, while used one set of stencils in the reconstruction, which is different from [13,15,16].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Zhao et al [25] developed a genuine fifth-order modified finite difference HWENO (M-HWENO) scheme in one and two dimensions. In [25], the authors modified the derivatives of the solution by a high-order Hermite limiter to control the derivatives near discontinuities and improve the stability of the scheme, while used one set of stencils in the reconstruction, which is different from [13,15,16]. Li et al [11] developed a multi-resolution HWENO scheme with unequal stencils, but the scheme only has the fourth-order accuracy in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we develop a simple fifth-order finite difference HWENO scheme combined with limiter (denoted as L-HWENO) for one-and two-dimensional hyperbolic conservation laws following the idea of the M-HWENO scheme [25]. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in [25], we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction.…”

Section: Introductionmentioning

confidence: 99%

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A Fifth-Order Finite Difference Hweno Scheme Combined with Limiter for Hyperbolic Conservation Laws

Zhang

Zhao²

2022

SSRN Journal

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“…For the spatial derivative of the point value v new i , we use the current point value u and the old spatial derivative v old to update it. We adopt a fourth order HWENO reconstruction proposed in [35]. We now roughly review the procedure of the reconstruction.…”

mentioning

confidence: 99%

High Order Residual Distribution Conservative Finite Difference HWENO Scheme for Steady State Problems

Lin,

Ren,

Abgrall

et al. 2021

Preprint

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In this paper, we develop a high order residual distribution (RD) method for solving steady state conservation laws in a novel Hermite weighted essentially non-oscillatory (HWENO) framework recently developed in [23]. In particular, we design a high order HWENO reconstructions for the integrals of source term and fluxes based on the point values of the solution and its spatial derivatives, and the principles of residual distribution schemes are adapted to obtain steady state solutions. The proposed novel HWENO framework enjoys two advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function, and compute them from the current value and the old spatial derivatives. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO framework. Second, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller at the same grid. Thus, it is also a compact scheme when we design the higher order accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical experiments for one and two-dimensional scalar and systems problems confirm the high order accuracy and good quality of our scheme.

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A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws

Cited by 18 publications

References 33 publications

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state $S_{N}$ transport equation

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state $S_{N}$ transport equation

A Fifth-Order Finite Difference Hweno Scheme Combined with Limiter for Hyperbolic Conservation Laws

High Order Residual Distribution Conservative Finite Difference HWENO Scheme for Steady State Problems

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