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Two-step Runge–Kutta methods for Volterra integro-differential equations

International Journal of Computer Mathematics 2024年第1期101卷 37-55页

作者： Wen, Jiao Huang, Chengming Guan, Hongbo College of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China

In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses. © 2024 Informa UK Limited, trading as Taylor & Francis Group.

关键词： Stability

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THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE

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Acta Mathematica Scientia 2020年第6期40卷 1928-1940页

作者： Wenxuan ZHENG Zhong TAN School of Mathematical Sciences Xiamen UniversityXiamen 361005China School of Mechanical and Electronic Engineering Tarim UniversityAlar 843300China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing Xiamen UniversityXiamen 361005China

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb *** spectral analysis and energy methods,we obtain the optimal time decay estimate of the *** show that the global classi... 详细信息

关键词： lower convergence rates upper decay rates spectral analysis energy method

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REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES

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Journal of Computational Mathematics 2023年第1期41卷 1-17页

作者： Haifeng Li Jing Zhang Jinming Wen Dongfang Li Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information ScienceHenan Normal UniversityXinxiang 453007China College of Information Science and Technology Jinan UniversityGuangzhou 510632China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China

In countless applications,we need to reconstruct a K-sparse signal x∈R n from noisy measurements y=Φx+v,whereΦ∈R^(m×n)is a sensing matrix and v∈R m is a noise *** least squares(OLS),which selects at each step the column that results in the most significant decrease in the residual power,is one of the most popular sparse recovery *** this paper,we investigate the number of iterations required for recovering x with the OLS *** show that OLS provides a stable reconstruction of all K-sparse signals x in[2.8K]iterations provided thatΦsatisfies the restricted isometry property(RIP).Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

关键词： Sparse signal recovery Orthogonal least squares(OLS) Restricted isometry property(RIP)

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A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

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Communications on Applied Mathematics and Computation 2020年第2期2卷 179-199页

作者： Junying Cao Ziqiang Wang Chuanju Xu School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

关键词： Caputo-Fabrizio derivative Fractional diferential equations High-order numerical scheme

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Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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Science china Mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

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Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

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Communications in Computational Physics 2018年第6期24卷 86-103页

作者： Dongfang Li Hong-Lin Liao Weiwei Sun Jilu Wang Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing211106China Department of Scientific Computing Florida StateUniversityTallahasseeFL 32306USA Beijing Computational Science Research Center Beijing 100094China.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element *** analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type *** this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional *** terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear *** theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

关键词： Time-fractional nonlinear parabolic problems L1-Galerkin FEMs Error estimates discrete fractional Gronwall type inequality Linearized schemes

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Dvelopment and application of a fully integrated model for unsaturated-saturated nitrogen reactive transport

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Agricultural Water Management 2017年 180卷 35-49页

作者： Zhu, Yan Yang, Jinzhong Ye, Ming Sun, Huaiwei Shi, Liangsheng State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University WuhanHubei430072 China Department of Scientific Computing Florida State University TallahasseeFL32306 United States School of Hydropower and Information Engineering Huazhong University of Science and Technology Wuhan430074 China

This paper presents a fully integrated model to simulate coupled unsaturated-saturated flow and reactive transport of ammonium and nitrate, the two major nitrogen species. Based on our previous work of developing a coupled model of unsaturated-saturated flow and solute transport of a single species, the key contribution of this study is to develop a new mathematical model and the new computer code for a comprehensive list of biogeochemical reactions, which are necessary for simulating nitrogen reactive transport. The computer code is verified by comparing its simulated results with those obtained using another model for a synthetic case. Model calibration and validation are conducted for two real-world cases to simulate nitrogen reactive transport. The two cases are at the experimental plot scale with the sewage water irrigation and at the field scale with the septic tank effluent infiltration, respectively. Quantitative evaluation of the numerical modeling results indicates that the new model and computer code can accurately simulate field observations of moisture content, water table elevation, and nitrogen concentration. For example, the root mean square error for simulating moisture content can be as small as 0.01. The code is computationally efficient to capture spatial and temporal trends of nitrogen concentrations at the field scale with a large number of computational nodes and time steps. The fully integrated model is a promising tool for large-scale modeling of water flow and nitrogen transformation and transport under complex conditions in agricultural and urban environments. © 2016 Elsevier B.V.

关键词： Mean square error

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Phase-field-based lattice Boltzmann method for containerless freezing

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Physical Review E 2024年第3期110卷 035301页

作者： Jiangxu Huang Lei Wang Zhenhua Chai Baochang Shi School of Mathematics and Statistics School of Mathematics and Physics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.

关键词： Drops & bubbles Liquid-solid phase transition

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Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations

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Physical Review E 2022年第5期106卷 055305-055305页

作者： Zhenhua Chai Baochang Shi Chengjie Zhan School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

关键词： Lattice-Boltzmann methods

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Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

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Physical Review E 2021年第3期104卷 035308-035308页

作者： Xinmeng Chen Zhenhua Chai Jinlong Shang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

关键词： Lattice-Boltzmann methods

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