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A LINEARLY-IMPLICIT structure-preserving EXPONENTIAL TIME DIFFERENCING SCHEME FOR HAMILTONIAN PDEs

Journal of Computational Mathematics 2024年第4期42卷 1063-1079页

作者： Yayun Fu Dongdong Hu Wenjun Cai Yushun Wang Henan Joint International Research Laboratory of High Performance Computation for Complex Systems School of ScienceXuchang UniversityXuchang 461000China School of Mathematics and Statistics Jiangxi Normal UniversityJiangxiNanchang 330022China Jiangsu Key Laboratory for NSLSCS Jiangsu Collaborative Innovation Center of Biomedial Functional MaterialsSchool of Mathematical SciencesNanjing Normal UniversityNanjing 210023China

In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear *** experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.

关键词： structure-preserving algorithm Hamiltonian PDE Energy quadratization method Exponential time differencing

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A LINEARLY-IMPLICIT ENERGY-preserving algorithm FOR THE TWO-DIMENSIONAL SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION BASED ON THE SAV APPROACH

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Journal of Computational Mathematics 2023年第5期41卷 797-816页

作者： Yayun Fu Wenjun Cai Yushun Wang School of Science Xuchang UniversityXuchang 461000China Jiangsu Key Laboratory for NSLSCS Jiangsu Collaborative Innovation Center of Biomedial Functional MaterialsSchool of Mathematical SciencesNanjing Normal UniversityNanjing 210023China

The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger ***,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar ***,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space *** that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete *** expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time ***,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

关键词： Fractional nonlinear Schrodinger equation Hamiltonian system Scalar auxiliary variable approach structure-preserving algorithm

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structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

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JOURNAL OF COMPUTATIONAL PHYSICS 2019年 395卷 166-185页

作者： Cai, Wenjun Jiang, Chaolong Wang, Yushun Song, Yongzhong Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing Jiangsu Peoples R China Yunnan Univ Finance & Econ Sch Stat & Math Kunming Yunnan Peoples R China

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the implicit midpoint method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions. (C) 2019 Elsevier Inc. All rights reserved.

关键词： structure-preserving algorithm Neumann boundary condition Summation by parts operator Scalar auxiliary variable approach

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A high-order structure-preserving difference scheme for generalized fractional Schrodinger equation with wave operator

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MATHEMATICS AND COMPUTERS IN SIMULATION 2023年第1期210卷 532-546页

作者： Zhang, Xi Ran, Maohua Liu, Yang Zhang, Li Sichuan Normal Univ Sch Math Sci Chengdu 610068 Peoples R China Sichuan Normal Univ VC & VR Key Lab Chengdu 610068 Peoples R China Aba Teachers Univ Sch Math Aba 623002 Peoples R China Sichuan Normal Univ VC & VR Key Lab Sichuan Prov Chengdu 610068 Peoples R China

This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrodinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.& COPY;2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Fractional Schrodinger equation with wave operator Scalar auxiliary variable approach structure-preserving algorithm Energy-conserving scheme Boundedness Convergence

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A DISSIPATION-preserving INTEGRATOR FOR DAMPED OSCILLATORY HAMILTONIAN SYSTEMS

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Journal of Computational Mathematics 2022年第4期40卷 570-588页

作者： Wei Shi Kai Liu College of Mathematical Sciences Nanjing Tech UniversityNanjing 211816China School of Statistics and Data Science Nanjing Audit UniversityNanjing 210023China

In this paper,based on discrete gradient,a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is *** solution of this system is a damped nonlinear ***,lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this *** new integrator gives a discrete analogue of the dissipation property of the original ***,since the integrator is based on the variation-of-constants formula for oscillatory systems,it preserves the oscillatory structure of the *** properties of the new integrator are *** convergence is analyzed for the implicit iterations based on the discrete gradient integrator,and it turns out that the convergence of the implicit iterations based on the new integrator is independent of k Mk,where M governs the main oscillation of the system and usually k Mk≫*** significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian *** experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature。

关键词： Weakly dissipative systems Oscillatory systems structure-preserving algorithm Discrete gradient integrator Sine-Gordon equation Continuousα-Fermi-Pasta-Ulam system

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A structure-preserving algorithm for the quaternion Cholesky decomposition

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APPLIED MATHEMATICS AND COMPUTATION 2013年 223卷 354-361页

作者： Wang, Minghui Ma, Wenhao Qingdao Univ Sci & Technol Dept Math Qingdao 266061 Peoples R China

In this paper, we discuss the Cholesky decomposition of the Hermitian positive definite quaternion matrix. For the first time, the structure-preserving Gauss transformation is defined, and then a novel structure-preserving algorithm, which is applied to its real representation matrix, is proposed. Our algorithm needs only real number operations, does not depend on the quaternion toolbox for matlab (QTFM) and has more portability. Although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or QTFM, numerical experiments show that our algorithm runs faster. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Quaternion matrix Cholesky decomposition structure-preserving algorithm

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CP DECOMPOSITION AND LOW-RANK APPROXIMATION OF ANTISYMMETRIC TENSORS

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2024年 62卷 72-94页

作者： Kovac, Erna Begovic Perisa, Lana Univ Zagreb Fac Chem Engn & Technol Marulicev Trg 19 Zagreb 10000 Croatia Visage Technol Ivana Lucica 2a Zagreb 10000 Croatia

For antisymmetric tensors, the paper examines a low-rank approximation that is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least-squares structure-preserving algorithm for finding such an approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-1 approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in the Julia programming language and their numerical performance is discussed.

关键词： CP decomposition antisymmetric tensors low-rank approximation structure-preserving algorithm Julia

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Efficient invariant-preserving scheme for the N-coupled nonlinear Schrödinger equations

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APPLIED MATHEMATICS LETTERS 2024年 157卷

作者： Cai, Jiaxiang Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China Minist Educ Key Lab NSLSCS Nanjing 210023 Jiangsu Peoples R China

The N -coupled nonlinear Schr & ouml;dinger equations are reformulated into an expanded form by using a scalar auxiliary variable method, and then a scheme preserving exactly original mass and energy conservation laws is proposed based on discretizing the expanded form. The scheme is efficient as it consists of decoupled linear systems with constant coefficients, along with a nonlinear algebraic equation that can be solved with negligible computational cost. Some numerical experiments are carried out to demonstrate the behavior of wave solutions, the accuracy of solution and the preservation of physical invariants.

关键词： Schr & ouml dinger equation structure-preserving algorithm Scalar auxiliary variable Energy-preserving algorithm Decoupled scheme

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Thermodynamically consistent hybrid computational models for fluid-particle interactions

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 513卷

作者： Hong, Qi Wang, Qi Nanjing Univ Aeronaut & Astronaut Sch Math Nanjing 211106 Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Air Nanjing 211106 Peoples R China Nanjing Normal Univ Key Lab Numer Simulat Large Scale Complex Syst Minist Educ Nanjing 210023 Peoples R China Univ South Carolina Dept Math Columbia SC 29208 USA

We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principle. This innovative framework harnesses the power of the phase -field -embedding method, in which each solid component, whether rigid or elastic, is characterized by a volume -preserving phase field. The unified velocity within the fluid -solid ensemble governs the movement of both solid particles and the surrounding fluid, specifically for passive particles. Active particles, however, are not only influenced by this unified velocity but are also driven by their self-propelling velocities. To capture exclusive volume interactions among particles and between particles and boundaries, we employ repulsive potential forces at a coarser scale. These forces effectively model repulsion and collision effects. Rigid particles maintain structural integrity by enforcing a zero velocity gradient tensor within their spatial domains, necessitating the introduction of a constraining stress tensor. In contrast, elastic particles are governed by a quasi -linear constitutive equation describing the elastic stress within their domains, allowing for accurate modeling of their deformations. The motion of solid particles is tracked by monitoring the dynamics of their centers of mass. This approach facilitates the development of a hybrid, thermodynamically consistent hydrodynamic model applicable to both rigid and elastic particles. To numerically solve this thermodynamically consistent model for elastic particles, we present a structure -preserving numerical algorithm. Notably, in the limit of an infinite elastic modulus, this algorithm converges to the one employed for modeling rigid particles. Finally, we substantiate the effectiveness, accuracy, and stability of our proposed scheme through a series of numerical experiments. These experiments not only validate the computati

关键词： Fluid-structure interaction Hydrodynamics Rigid and elastic body Passive and active matter Phase field structure-preserving algorithm

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An efficient energy-preserving method for the two-dimensional fractional Schrodinger equation

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APPLIED NUMERICAL MATHEMATICS 2021年 165卷 232-247页

作者： Fu, Yayun Xu, Zhuangzhi Cai, Wenjun Wang, Yushun Xuchang Univ Sch Sci Xuchang 461000 Peoples R China Nanjing Normal Univ Jiangsu Collaborat Innovat Ctr Biomedial Funct Ma Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrodinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Fractional nonlinear Schrodinger equation Hamiltonian structure structure-preserving algorithm Exponential time differencing

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