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EXPONENTIAL INTEGRATORS preserving FIRST INTEGRALS OR LYAPUNOV FUNCTIONS FOR CONSERVATIVE OR DISSIPATIVE SYSTEMS

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第3期38卷 A1876-A1895页

作者： Li, Yu-Wen Wu, Xinyuan Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ State Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China

In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system (y) over dot = Q(My vertical bar del U(y)), where Q is a dxd skew-symmetric or negative semidefinite real matrix, M is a dxd symmetric real matrix, and U : R-d -> R is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.

关键词： first integral Lyapunov function variation-of-constants formula exponential integrator discrete gradient structure-preserving algorithm

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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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A fast structure-preserving method for computing the singular value decomposition of quaternion matrices

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APPLIED MATHEMATICS AND COMPUTATION 2014年 235卷 157-167页

作者： Li, Ying Wei, Musheng Zhang, Fengxia Zhao, Jianli Liaocheng Univ Coll Math Sci Shandong 252000 Peoples R China Shanghai Normal Univ Coll Math & Sci Shanghai 200234 Peoples R China

In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structure-preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Quaternion matrix Quaternion singular value decomposition structure-preserving algorithm

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A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach

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APPLIED NUMERICAL MATHEMATICS 2021年 160卷 368-385页

作者： Fu, Yayun 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

This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we utilize the fractional centered difference formula to discrete the equivalent system derived by the invariant energy quadratization method in space direction, and obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully-discrete conservative scheme. The stability, solvability and convergence in the maximum norm of the numerical scheme are given. Furthermore, a fast algorithm based on the fast Fourier transformation technique is used to reduce the computational complexity in practical computation. Finally, numerical examples are provided to confirm our theoretical analysis results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： structure-preserving algorithm Fractional sine-Gordon equation Hamiltonian system Invariant energy quadratization Numerical analysis

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structure-preserving EXPONENTIAL RUNGE-KUTTA METHODS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第2期39卷 A593-A612页

作者： Bhatt, Ashish Moore, Brian E. Univ Cent Florida Dept Math Orlando FL 32816 USA

Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) methods are developed for solving initial value problems with vector fields that can be split into conservative and linear nonconservative parts. The focus is on linearly damped ordinary differential equations that possess certain invariants when the damping coefficient is zero, but, in the presence of constant or time-dependent linear damping, the invariants satisfy linear differential equations. Similar to the way that Runge-Kutta and partitioned Runge-Kutta methods preserve quadratic invariants and symplecticity for Hamiltonian systems, ERK and PERK methods exactly preserve conformal symplecticity, as well as decay (or growth) rates in linear and quadratic invariants, under certain constraints on their coefficient functions. Numerical experiments illustrate the higher-order accuracy and structure-preserving properties of various ERK methods, demonstrating clear advantages over classical conservative Runge-Kutta methods, as well as usefulness for solving a wide range of differential equations.

关键词： structure-preserving algorithm conformal symplectic time-dependent damping exponential integrators integrating factor methods exponential time differencing

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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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Efficient energy-preserving integrators for oscillatory Hamiltonian systems

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年 235卷 587-605页

作者： Wu, Xinyuan Wang, Bin Shi, Wei Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ State Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China

In this paper, we focus our attention on deriving and analyzing an efficient energy-preserving formula for the system of nonlinear oscillatory or highly oscillatory second-order differential equations q ''(t) + Mq(t) - f(q(t)), where M is a symmetric positive semi-definite matrix with parallel to M parallel to >> 1 and f(q) = -del U-q(q) is the negative gradient of a real-valued function U(q). This system is a Hamiltonian system with the Hamiltonian H(p, q) = 1/2p(T)p + 1/2q(T)Mq+U(q). The energy-preserving formula exactly preserves the Hamiltonian. We analyze in detail the properties of the energy-preserving formula and propose new efficient energy-preserving integrators in the sense of numerical implementation. The convergence analysis of the fixed-point iteration is presented for the implicit integrators proposed in this paper. It is shown that the convergence of implicit Average Vector Field methods is dependent on parallel to M parallel to, whereas the convergence of the new energy-preserving integrators is independent of parallel to M parallel to. The Fermi-Pasta-Ulam problem and the sine-Gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well-known Average Vector Field methods in the scientific literature. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Hamiltonian system Energy-preserving formula structure-preserving algorithm Oscillatory differential equation Average Vector Field formula Fermi-Pasta-Ulam problem Sine-Gordon equation

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An SDG Galerkin structure-preserving scheme for the Klein-Gordon-Schrodinger equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2020年第9期43卷 6011-6030页

作者： Wang, Jialing Wang, Yushun Nanjing Univ Informat Sci & Technol Sch Math & Stat Nanjing 210044 Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

In this paper, we use the Galerkin weak form to construct a structure-preserving scheme for Klein-Gordon-Schrodinger equation and analyze its conservative and convergent properties. We first discretize the underlying equation in space direction via a selected finite element method, and the Hamiltonian partial differential equation can be casted into Hamiltonian ordinary differential equations based on the weak form of the system afterwards. Then, the resulted ordinary differential equations are solved by the symmetric discrete gradient method, which yields a charge-preserving and energy-preserving scheme. Moreover, the numerical solution of the proposed scheme is proved to be bounded in the discrete L infinity norm and convergent with the convergence order of O(h2+tau 2) in the discrete L2 norm without any grid ratio restrictions, where h and tau are space and time step, respectively. Numerical experiments conducted last to verify the theoretical analysis.

关键词： finite element method Klein-Gordon-Schrodinger equation structure-preserving algorithm symmetric discrete gradient method

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Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations

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MATHEMATICS AND COMPUTERS IN SIMULATION 2009年第1期80卷 20-28页

作者： Moore, Brian E. Univ Cent Florida Dept Math Orlando FL 32816 USA

Conformal symplecticity is generalized to forced-damped multi-symplectic PDEs in 1 + 1 dimensions Since,I conformal multi-symplectic property has a concise form for these equations. numerical algorithms that preserve this property. from a modified equations point of view. are available. In effect. the modified equations for standard multi-symplectic methods and for space-time splitting methods satisfy a conformal multi-symplectic property, and the splitting schemes exactly preserve global symplecticity in a special case. It is also shown that the splitting schemes yield incorrect rates of energy/momentum dissipation, but this IS not the case for standard multi-symplectic schemes. These methods work best. for problems where the dissipation coefficients are small, and a forced-damped semi-linear wave equation is considered as an example Published by Elsevier B.V. on behalf of IMACS.

关键词： Multi-symplectic PDE Conformal symplectic structure-preserving algorithm Splitting methods Modified equations

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A new structure-preserving method for quaternion Hermitian eigenvalue problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年第1期239卷 12-24页

作者： Jia, Zhigang Wei, Musheng Ling, Sitao Shanghai Normal Univ Coll Math & Sci Shanghai 200234 Peoples R China Jiangsu Normal Univ Sch Math Sci Jiangsu 221116 Peoples R China China Univ Min & Technol Coll Sci Jiangsu 221116 Peoples R China

In this paper we propose a novel structure-preserving algorithm for solving the right eigenvalue problem of quaternion Hermitian matrices. The algorithm is based on the structure-preserving tridiagonalization of the real counterpart for quaternion Hermitian matrices by applying orthogonal JRS-symplectic matrices. The algorithm is numerically stable because we use orthogonal transformations;the algorithm is very efficient, it costs about a quarter arithmetical operations, and a quarter to one-eighth CPU times, comparing with standard general-purpose algorithms. Numerical experiments are provided to demonstrate the efficiency of the structure-preserving algorithm. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Quaternion Hermitian operator Quaternionic right eigenvalue problem structure-preserving algorithm

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