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An oscillation-free Hermite WENO scheme for hyperbolic conservation laws

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Science China Mathematics 2024年第2期67卷 431-454页

作者： Zhuang Zhao jianxian qiu School of Mathematical Sciences and Institute of Natural Sciences Shanghai Jiao Tong UniversityShanghai 200240China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OFHWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- andfirst-order moments are the variables for the governing equations. The main difference from other HWENOschemes existing in the literature is that we add high-order numerical damping terms in the first-order momentequations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only canachieve the designed optimal numerical order, but also can be easily implemented as we use only one set ofstencils in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- andfirst-order moment equations. In order to obtain the adaptive order resolution when facing discontinuities, atransition polynomial is added in the reconstruction, where the associated linear weights can also be any positivenumbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps compactnessas only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests areperformed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme.

关键词： Hermite WENO scheme hyperbolic conservation laws oscillation-free adaptive order discontinuous Galerkin method

Trigonometric WENO Schemes for Hyperbolic Conservation Laws and Highly Oscillatory Problems

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Communications in Computational Physics 2010年第10期8卷 1242-1263页

作者： Jun Zhu jianxian qiu College of Science Nanjing University of Aeronautics and AstronauticsNanjingJiangsu 210016China Department of Mathematics Nanjing UniversityNanjingJiangsu 210093China

In this paper,we use trigonometric polynomial reconstruction,instead of algebraic polynomial reconstruction,as building blocks for the weighted essentially non-oscillatory(WENO)finite difference schemes to solve hyperbolic conservation laws and highly oscillatory *** goal is to obtain robust and high order accurate solutions in smooth regions,and sharp and non-oscillatory shock *** results are provided to illustrate the behavior of the proposed schemes.

关键词： TWENO scheme hyperbolic conservation laws highly oscillatory problem finite difference scheme

Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

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Communications on Applied Mathematics and Computation 2024年第1期6卷 605-624页

作者： Feng Zheng jianxian qiu College of Mathematics and Statistics Fujian Normal UniversityFuzhou350117FujianChina School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling&High-Performance Scientific Computing Xiamen UniversityXiamen361005FujianChina

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation *** can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary ***,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO *** numerical tests are performed to verify the high resolution and high accuracy of the scheme.

关键词： Finite volume Dimension by dimension HWENO Hyperbolic conservation laws

New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

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Communications on Applied Mathematics and Computation 2023年第1期5卷 64-96页

作者： Jun Zhu jianxian qiu College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen 361005FujianChina

In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation *** the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more *** more we use wider candidate stencils,the bigger the probability of discontinuities lies in all ***,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point *** the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(***.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth *** new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL *** numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

关键词： Finite difference Mapped WENO scheme Mapping function Mapped nonlinear weight Unequal-sized stencil Extreme example

A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes

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Science China Mathematics 2008年第8期51卷 1549-1560页

作者： ZHU Jun qiu jianxian Department of Mathematics Nanjing UniversityNanjing 210093China College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016China

In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation *** key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional *** main difference between this work and the formal one is the procedure to reconstruct the derivative *** with the original HWENO schemes of qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong *** numerical experiments are performed to illustrate the capability of the method.

关键词： finite volume HWENO scheme conservation laws Hermite polynomial TVD Runge-Kutta time discretization method 65M06 65M99 35L65

Development and Comparison of Numerical Fluxes for LWDG Methods

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Numerical Mathematics(Theory,Methods and Applications) 2008年第4期1卷 435-459页

作者： jianxian qiu Department of Mathematics Nanjing University Nanjing Jiangsu 210093 China.

The discontinuous Galerkin （DO） or local discontinuous Galerkin （LDG） method is a spatial discretization procedure for convection-diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax- Wendroff time discretization procedure is an altemative method for time discretization to the popular total variation diminishing （TVD） Runge-Kutta time discretizations. In this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure （LWDG） based on different numerical fluxes for finite volume or finite difference schemes, including the first-order monotone fluxes such as the Lax-Friedfichs flux, Godunov flux, the Engquist-Osher flux etc. and the second-order TVD fluxes. We systematically investigate the performance of the LWDG methods based on these different numerical fluxes for convection terms with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

关键词： Discontinuous Galerkin method Lax-Wendroff type time discretization numerical flux approximate Riemann solver timiter WENO scheme high order accuracy.

HIGH ORDER FINITE DIFFERENCE HERMITE WENO FAST SWEEPING METHODS FOR STATIC HAMILTON-JACOBI EQUATIONS

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Journal of Computational Mathematics 2023年第6期41卷 1064-1092页

作者： Yupeng Ren Yulong Xing jianxian qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005China Department of Mathematics The Ohio State UniversityColumbusOH 43210USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005China

In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations *** the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small ***,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update *** traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare *** novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO *** addition,a hybrid strategy is also introduced to further reduce computational *** numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.

关键词： Finite difference Hermite methods Weighted essentially non-oscillatory method Fast sweeping method Static Hamilton-Jacobi equations Eikonal equation

Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters:Three-Dimensional

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Communications in Computational Physics 2012年第3期11卷 985-1005页

作者： Jun Zhu jianxian qiu College of Science Nanjing University of Aeronautics and AstronauticsNanjingJiangsu 210016P.R.China School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China and Department of MathematicsNanjing UniversityNanjingJiangsu 210093P.R.China

This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation *** application discussed here is the solution of 3-D problems on unstructured *** numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.

关键词： Runge-Kutta discontinuous Galerkin method limiter WENO HWENO high order limiting procedure

An h-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

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Numerical Mathematics(Theory,Methods and Applications) 2013年第4期6卷 617-636页

作者： Hongqiang Zhu jianxian qiu School of Natural Science Nanjing University of Posts and TelecommunicationsNanjing 210023JiangsuChina School of Mathematical Sciences Xiamen UniversityXiamen 361005FujianChina

In[35,36],we presented an h-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws.A tree data structure(binary tree in one dimension and quadtree in two dimensions)is used to aid storage and neighbor *** adaptation is achieved by refining the troubled cells and coarsening the untroubled"children".Extensive numerical tests indicate that the proposed h-adaptive method is capable of saving the computational cost and enhancing the resolution near the *** this paper,we apply this h-adaptive method to solve Hamilton-Jacobi equations,with an objective of enhancing the resolution near the discontinuities of the solution ***-and two-dimensional numerical examples are shown to illustrate the capability of the method.

关键词： Runge-Kutta discontinuous Galerkin method h-adaptive method Hamilton-Jacobi equation