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arXiv 2024年

作者： Du, Kui Fan, Jia-Jun Wang, Fang School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mathematical Sciences Xiamen University Xiamen361005 China

We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed *** Codes 15A06, 65F10, 65F25, 65F50 Copyright © 2024, The Authors. All rights reserved.

关键词： Linear systems

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Adaptive OrderWENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

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Communications in Computational Physics 2021年第6期30卷 67-96页

作者： Jiajie Chen Xiaofeng Cai Jianxian Qiu Jing-Mei Qiu Department of Mathematical Sciences University of DelawareNewarkDE19717USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive *** is an extension of the conservative semi-Lagrangian(SL)finite difference WENO scheme in[Qiu and Shu,JCP,230(4)(2011),pp.863-889],in which linear weights in SL WENO framework were shown to not exist for variable coefficient ***,the order of accuracy is not optimal from reconstruction *** this paper,we incorporate a recent WENO adaptive order(AO)technique[Balsara et al.,JCP,326(2016),pp.780-804]to the SL WENO *** new scheme can achieve an optimal high order of accuracy,while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL *** positivity-preserving limiter is further applied to ensure the positivity of ***,the scheme is applied to high dimensional problems by a fourth-order dimensional *** demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations,the Vlasov-Poisson system,the guiding center Vlasov model as well as the incompressible Euler equations.

关键词： Semi-Lagrangian weighted essentially nonoscillatory WENO adaptive order reconstruction finite difference mass conservation Vlasov-Poisson incompressible Euler

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High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

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Communications in Computational Physics 2022年第1期31卷 154-187页

作者： Yupeng Ren Yulong Xing Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China 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 361005P.R.China

In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any HamiltonJacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.

关键词： Hermite method Weighted Essentially Non-Oscillatory(WENO)method fixedpoint iteration Hamilton-Jacobi equation hybrid strategy

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Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

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Communications in Computational Physics 2022年第7期32卷 364-400页

作者： Jiayin Li Chi-Wang Shu Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathe-matical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation *** main idea of this scheme is derived from our previous work[***.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the ***,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order *** addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near *** obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO ***,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO *** numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

关键词： Moment-based scheme multi-resolution scheme HWENO scheme hyperbolic conservation laws KXRCF troubled-cell indicator HLLC-flux

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STABILITY AND ERROR ANALYSIS OF PRESSURE-CORRECTION SCHEME FOR THE NAVIER-STOKES-PLANCK-NERNST-POISSON EQUATIONS

arXiv

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arXiv 2024年

作者： He, Yuyu Chen, Hongtao School of Mathematical Sciences Xiamen University Xiamen Fujian361005 China Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen Fujian361005 China

In this paper, we propose and analyze first-order time-stepping pressure-correction projection scheme for the Navier-Stokes-Planck-Nernst-Poisson equations. By introducing a governing equation for the auxiliary variable through the ionic concentration equations, we reconstruct the original equations into an equivalent system and develop a first-order decoupled and linearized scheme. This scheme preserves non-negativity and mass conservation of the concentration components and is unconditionally energy stable. We derive the rigorous error estimates in the two dimensional case for the ionic concentrations, electric potential, velocity and pressure in the L2- and H1-norms. Numerical examples are presented to validate the proposed *** Codes 65M12, 65M15, 65N30, 76M10 © 2024, CC BY.

关键词： Navier Stokes equations

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HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

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Journal of Computational Mathematics 2020年第4期38卷 580-605页

作者： Mohammad Tanzil Hasan Chuanju Xu School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper,efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water *** using the Caputo fractional derivative definition to approximate the nonlocal fractional operator,finite difference method in time and spectral method in space are constructed for the considered *** proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear *** analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time,and spectral accuracy in *** numerical results are provided to verify the efficiency and accuracy of our theoretical ***,the decay rate of solutions are investigated.

关键词： Water waves Nonlocal viscosity Finite difference Spectral method Conver-gence order Decay rate

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AN EFFICIENT SPECTRAL TRUST-REGION DEFLATION METHOD FOR MULTIPLE SOLUTIONS

arXiv

引用

arXiv 2023年

作者： Li, Lin Wang, Li-Lian Li, Huiyuan School of Mathematics and Physics Hunan Key Laboratory for Mathematical Modeling and Scientific Computing University of South China Hengyang421001 China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore State Key Laboratory of Computer Science Laboratory of Parallel Computing Institute of Software Chinese Academy of Sciences Beijing100190 China

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions;(ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS;and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions;(ii) it guarantees high-order accuracy;and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature. Copyright © 2023, The Authors. All rights reserved.

关键词： Newton-Raphson method

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Inverse Lax-Wendroff boundary treatment for solving conservation laws with finite difference HWENO methods

arXiv

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arXiv 2025年

作者： Zhu, Guanghao Jiang, Yan Zhao, Zhuang Zhang, Mengping School of Mathematical Sciences University of Science and Technology of China Anhui Hefei230026 China Laoshan Laboratory Qingdao266237 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

This paper presents a novel inverse Lax-Wendroff (ILW) boundary treatment for finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes to solve hyperbolic conservation laws on arbitrary geometries. The complex geometric domain is divided by a uniform Cartesian grid, resulting in challenge in boundary treatment. The proposed ILW boundary treatment could provide high order approximations of both solution values and spatial derivatives at ghost points outside the computational domain. Distinct from existing ILW approaches, our boundary treatment constructs the extrapolation via optimized through a least squares formulation, coupled with the spatial derivatives at the boundary obtained via the ILW procedure. Theoretical analysis indicates that compared with other ILW methods, our proposed one would require fewer terms by using the relatively complicated ILW procedure and thus improve computational efficiency while preserving accuracy and stability. The effectiveness and robustness of the method are validated through numerical experiments. Copyright © 2025, The Authors. All rights reserved.

关键词： Numerical methods

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Large contributions from optical phonons to thermal transport in hexagonal carbon-boron-nitrogen monolayers

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Physical Review B 2023年第23期108卷 235420-235420页

作者： Xue-Kun Chen (陈学坤) Yue Zhang (张越) Qing-Qing Luo (罗青清) Xue Chen (陈雪) Pin-Zhen Jia (贾聘真) Wu-Xing Zhou (周五星) School of Mathematics and Physics and Hunan Key Laboratory of Mathematical Modeling and Scientific Computing University of South China Hengyang 421001 China Department of Mathematics and Physics Hunan Institute of Technology Hengyang 421002 China School of Materials Science and Engineering and Hunan Provincial Key Laboratory of Advanced Materials for New Energy Storage and Conversion Hunan University of Science and Technology Xiangtan 411201 People's Republic of China

As representative semiconducting hexagonal carbon-boron-nitride lattices, C6BN and C2BN are experimentally realized two-dimensional (2D) plane materials and have recently become the focus of research. Herein, combining first-principles calculations with the Boltzmann transport equation, we performed a comprehensive study on the phonon interaction and thermal conductivity in C6BN and C2BN monolayers. It is found that the thermal conductivities of C6BN and C2BN monolayers at room temperature are reduced by 79% and 73%, respectively, due to four-phonon scattering, compared with the results including three-phonon scattering only. We can attribute this phenomenon to giant four-phonon scattering exclusive for the heat-carrying out-of-plane acoustic (ZA) phonons, because the reflection symmetry allows four-ZA processes much higher than three-ZA processes, and the quasiparallel behavior between the ZA and low-lying out-of-plane optical (ZO) branches contribute to a broad phase space for four-phonon scattering as well. Moreover, C6BN monolayer exhibits unusual behavior that optical phonons contribute about ∼60% to the overall thermal conductivity under the four-phonon picture, which differs from the traditional case that acoustic phonons dominate thermal conductivity. Unexpectedly, two low-lying ZO modes have as high as 38% contributions to the thermal transport at 300 K under the four-phonon picture, causing 60% contribution of optical phonon modes, apparently larger than that of the three-phonon case (15%) and many other 2D materials, also indicating the four-phonon scattering has a more significant effect on acoustic phonons than on optical phonons. This finding not only highlights insight into the nature of phonon transport, but also provides a promising strategy for manipulation of heat transport based on optical phonon modes.

关键词： Phonons Thermal conductivity Boron nitride Graphene First-principles calculations Lattice-Boltzmann methods

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High Order Asymptotic Preserving Finite Difference Weno Schemes with Constrained Transport for Mhd Equations in All Sonic Mach Numbers

SSRN

引用

SSRN 2023年

作者： Chen, Wei Wu, Kailiang Xiong, Tao School of Mathematical Sciences Xiamen University Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Fujian Xiamen361005 China Guangdong Shenzhen518055 China

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergencefree finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number Ε ranging from 0 to O(1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially nonoscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit. © 2023, The Authors. All rights reserved.

关键词： Mach number

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