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High Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows

Communications in Computational Physics 2022年第10期32卷 1474-1509页

作者： Nan Zhang Zhiping Mao Tao Xiong School of Mathematical Sciences Xiamen UniversityXiamenFujian361005P.R.China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenFujian361005P.R.China

In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential *** fractional Laplacian is approximated by a Fourier-Galerkin spectral method,while the time evolution of the potential temperature is discretized by a high order conservative finite difference *** essentially non-oscillatory(WENO)reconstructions are also considered for *** to a low regularity of problems involving such a fractional Laplacian,especially in the critical or supercritical regime,directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity,*** smooth *** of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference *** results are performed to demonstrate the good performance of our proposed approach.

关键词： Finite difference WENO Fourier spectral fractional Laplacian surface quasigeostrophic flow

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A High Order Bound Preserving Finite Difference Linear Scheme for Incompressible Flows

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Communications in Computational Physics 2022年第6期32卷 126-155页

作者： Guoliang Zhang Tao Xiong School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

We propose a high order finite difference linear scheme combined with ahigh order bound preserving maximum-principle-preserving (MPP) flux limiter tosolve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much lesscostly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP fluxlimiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guidingcenter model and the incompressible Euler system. The comparison between the linearand WENO type schemes, with and without the MPP flux limiter, will demonstrate thegood performance of our proposed approach.

关键词： Finite difference scheme high order Hermite reconstruction MPP flux limiter incompressible flow Vlasov-Poisson

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Gravitational lensing of massive particles by a black-bounce-Schwarzschild black hole

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Physical Review D 2024年第6期110卷 064008页

作者： Guansheng He Yi Xie Chunhua Jiang Wenbin Lin School of Mathematics and Physics Chinese Academy of Sciences Nanjing 210023 China Hunan Key Laboratory of Mathematical Modeling and Scientific Computing Hengyang 421001 China School of Physical Science and Technology

We investigate in detail the weak-field gravitational lensing of a relativistic neutral massive particle induced by a regular black-bounce-Schwarzschild black hole proposed by Simpson and Visser. Starting with the calculation of the gravitational deflection of the massive particle up to the third post-Minkowskian order, the Virbhadra-Ellis lens equation is solved perturbatively beyond the weak-deflection limit to achieve the expressions for the lensing observables of the primary and secondary images of a pointlike particle source. The main observables contain not only the positions, the flux magnifications, and the gravitational time delays of the individual images, but also the positional relations, the magnification relations (including the total magnification), the magnification centroid, and the differential time delay. We then discuss the velocity-induced effects originated from the deviation of the particle’s initial velocity from the speed of light on the black-bounce-Schwarzschild lensing observables of the images of a pointlike light source and the effects induced by the bounce parameter of the spacetime on the measurable image properties of Schwarzschild lensing of the massive particle. As an application of the results, we model the supermassive black hole in the Galactic Center (i.e., Sgr A*) as the lens and focus on evaluating the possibilities to detect the new velocity- and bounce-induced effects on the practical lensing observables and analyzing the dependence of these effects on the parameters.

关键词： General relativity Gravitational lenses

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THE TIME DECAY RATES OF THE CLASSICAL SOLUTION TO THE POISSON-NERNST-PLANCK-FOURIER EQUATIONS IN R^(3)

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Acta Mathematica Scientia 2022年第3期42卷 1081-1102页

作者： Leilei TONG Zhong TAN Xu ZHANG School of Science Chongqing University of Posts and TelecommunicationsChongqing 400065China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling&High Performance Scientific Computing Xiamen UniversityXiamen 361005China School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

In this work,the Poisson-Nernst-Planck-Fourier system in three dimensions is *** when the initial data regards a small perturbation around the constant equilibrium state in a H^(3)∩■^(-s)(0≤s≤1/2)norm,we obtain th... 详细信息

关键词： Poisson-Nernst-Planck-Fourier system decay rates energy method

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Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System

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Communications on Applied Mathematics and Computation 2022年第1期4卷 34-59页

作者： Francis Filbet Tao Xiong Institut de Mathématiques de Toulouse UniversitéPaul SabatierToulouse 31062France School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen 361005Fujian ProvinceChina

We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity *** schemes are designed to be systematically as accurate as one wants with prov-able conservation of mass and possibly total *** properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson *** proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations,resulting in a consistent description of the distribu-tion function and the electric *** simulations are performed to verify the order of the accuracy and conservation properties.

关键词： Energy conserving Discontinuous Galerkin method Hermite spectral method Vlasov-Poisson

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A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

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Communications in Computational Physics 2022年第1期31卷 94-130页

作者： Min Zhang Weizhang Huang Jianxian Qiu School of Mathematical Sciences Peking UniversityBeijing100871China Department of Mathematics University of KansasLawrenceKansas 66045USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.

关键词： Well-balance positivity-preserving high-order accuracy quasi-Lagrange moving mesh DG method shallow water equations

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Adaptive fully implicit and thermodynamically consistent modeling of multiphase flow in porous media on three dimensional grids

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Computers and Fluids 2025年 298卷

作者： Sheng, Qiuyu Yang, Haijian Chen, Huangxin Cheng, Tianpei Sun, Shuyu School of Mathematics Hunan University Changsha Hunan410082 China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Tongji University Shanghai200092 China

The development of innovative mathematical models and state-of-the-art simulators for multiphase flow in porous media is a key focus in hydrogeology. However, many traditional models for multiphase flow in porous media lack complete thermodynamic consistency, as the energy reconstructed from the capillary pressure fails to yield a time-dependent system with a dissipated free energy. In this paper, we utilize a thermodynamically consistent model for multiphase flow in porous media, which satisfies the second law of thermodynamics. For large-scale numerical simulation of the resultant flow model, we introduce and investigate a robust and scalable fully implicit simulator with a suitable time adaptivity strategy designed for distributed memory parallel computers. In particular, our approach enhances the numerical formulation by proposing a family of inexact Newton–Krylov methods for efficient computation and several types of field-split algorithms for large-scale preconditioning. The numerical experiments indicate that the proposed fully implicit simulator accurately predicts the highly complex physical processes of thermodynamically consistent problems, particularly those characterized by high heterogeneity with complex reservoir topography on 3D structured or unstructured grids. © 2025 Elsevier Ltd

关键词： Parallel processing systems

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EFFICIENT NUMERICAL METHODS FOR THE NAVIER-STOKES-NERNST-PLANCK-POISSON EQUATIONS

arXiv

引用

arXiv 2023年

作者： Zhou, Xiaolan Xu, Chuanju School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China

We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable reformulation and sophisticated treatment of the terms coupling different equations. By introducing a dynamic equation for the auxiliary variable and reformulating the original equations into an equivalent system, we construct first- and second-order semi-implicit linearized schemes for the underlying problem. The main advantages of the proposed method are: (1) the schemes are unconditionally stable in the sense that a discrete energy keeps decay during the time stepping;(2) the concentration components of the discrete solution preserve positivity and mass conservation;(3) the delicate implementation shows that the proposed schemes can be very efficiently realized, with computational complexity close to a semi-implicit scheme. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method. As far as the best we know, this is the first second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations. © 2023, CC BY.

关键词： Numerical methods

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A NONLINEAR LEAST-SQUARES CONVEXITY ENFORCING C⁰ INTERIOR PENALTY METHOD FOR THE MONGE–AMPÈRE EQUATION ON STRICTLY CONVEX SMOOTH PLANAR DOMAINS

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Communications of the American mathematical Society 2024年第14期4卷 607-640页

作者： Brenner, Susanne C. Sung, Li-Yeng Tan, Zhiyu Zhang, Hongchao Department of Mathematics and Center for Computation and Technology Louisiana State University Baton Rouge 70803 LA United States Center for Computation and Technology Louisiana State University Baton Rouge LA United States School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian 361005 China

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R². It is based on an isoparametric C⁰ finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented. © 2024 by the author(s) under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License (CC BY NC ND 4.0)

关键词： Monge-Ampère equation strictly convex smooth domains nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method enhanced cubic Lagrange finite elements isoparametric a priori and a posteriori error estimates active set algorithm

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Obtaining the pseudoinverse solution of singular range-symmetric linear systems with GMRES-type methods

arXiv

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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

It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum A-residual) for range-symmetric linear systems Ax = b. At step k RSMAR minimizes Ark in the kth Krylov subspace generated with {A, r0} rather than rk, where rk is the kth residual vector and · denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear *** Codes 15A06, 15A09, 65F10, 65F25, 65F50 Copyright © 2024, The Authors. All rights reserved.

关键词： Iterative methods

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