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Local energy-preserving algorithms for nonlinear fourth-order Schrodinger equation with trapped term

APPLIED MATHEMATICS AND COMPUTATION 2017年 296卷 23-32页

作者： Cai, Jiaxiang Liang, Hua Yang, Bin Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China

Based on the rule that numerical algorithms should preserve the intrinsic properties of the original problem as many as possible, we propose two local energy-preserving algorithms for the nonlinear fourth-order Schrodinger equation with a trapped term. The local energy conservation law is preserved on any local time-space region. With appropriate boundary conditions, the first algorithm will be both globally charge- and energy-preserving and the second one will be energy-preserving. Numerical experiments show that the proposed algorithms provide more accurate solution than many existing methods and also exhibit excellent performance in preserving conservation laws. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Schrodinger equation structure-preserving algorithm Local property Conservation law Energy

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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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Multi-conformal-symplectic PDEs and discretizations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 323卷 1-15页

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

Past work on integration methods that preserve a conformal symplectic structure focuses on Hamiltonian systems with weak linear damping. In this work, systems of PDEs that have conformal symplectic structure in time and space are considered, meaning conformal symplecticity is fully generalized for PDEs. Using multiple examples, it is shown that PDEs with this particular structure have interesting applications. What it means to preserve a multi-conformal-symplectic conseivation law numerically is explained, along with presentation of two numerical methods that preserve such properties. Then, the advantages of the methods are briefly explored through applications to linear equations, consideration of momentum and energy dissipation, and backward error analysis. Numerical simulations for two PDEs illustrate the properties of the methods, as well as the advantages over other standard methods. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Multi-symplectic PDE structure-preserving algorithm Conformal symplectic

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High-order integration scheme for relativistic charged particle motion in magnetized plasmas with volume preserving properties

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COMPUTER PHYSICS COMMUNICATIONS 2017年 220卷 285-296页

作者： Matsuyama, Akinobu Furukawa, Masaru Natl Inst Quantum & Radiol Sci & Technol Rokkasho Aomori 0393212 Japan Tottori Univ Grad Sch Engn Tottori Tottori 6808552 Japan

A numerical algorithm for solving full gyro orbit of relativistic charged particle motion in magnetized plasmas is presented. The algorithm developed here achieves the following features simultaneously. (1) The time advancement is explicit, and (2) the integration is performed with respect to the observation time in a laboratory frame. (3) It is suitable for accurate long time integration with its volume preserving properties in the phase space. (4) The algorithm can properly treat the E x B drift velocity in electromagnetic fields for large relativistic factors, and (5) can be extended to arbitrary high orders with the aid of symmetric composition methods. Because our algorithm is formulated in the Lorentz covariant form, explicit conservation of the Minkowski norm is not assumed. Nevertheless, no secular growth of the numerical errors in the norm does occur, and its influence can be minimized up to the levels of round-off errors when a high-order scheme is applied. Numerical results are compared with explicit and implicit Runge-Kutta methods. The numerical accuracy and the computational efficiency are discussed for long time integration in toroidal magnetic field configuration. (C) 2017 Elsevier B.V. All rights reserved.

关键词： structure-preserving algorithm Charged particle motion Hamiltonian mechanics Tokamaks

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A LINEARLY-FITTED CONSERVATIVE （DISSIPATIVE） SCHEME FOR EFFICIENTLY SOLVING CONSERVATIVE （DISSIPATIVE） NONLINEAR WAVE PDES

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Journal of Computational Mathematics 2017年第6期35卷 780-800页

作者： Kai Liu XinyuanWu Wei Shi College of Applied Mathematics Nanjing University of Finance ~ Economics Nanjing 210023 China Department of Mathematics Nanjing University State Key Laboratory for Novel Software Technology at Nanjing University Nanjing 210093 China College of Mathematical Sciences Nanjing Teeh University Nanjing 211816 China

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative （dissipative） nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

关键词： Conservative （dissipative） wave PDEs structure-preserving algorithm Linearly-fitted Average Vector Field formula Sine-Gordon equation.

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

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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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Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第3期66卷 1234-1259页

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

Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Stormer-Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissipation of angular momentum of an N-body system with pairwise distance dependent interactions. Necessary and sufficient conditions for second order accuracy are derived. Analysis for linear equations gives explicit relationships between the damping parameter and the step size to reveal when the methods are most advantageous;essentially, the damping rate of the numerical solution is exactly preserved under these conditions. The methods are applied to several model problems, both ODEs and PDEs. Additional structure preservation is discovered for the discretized PDEs, in one case dissipation in total linear momentum and in another dissipation in mass are preserved by the methods. The numerical results, along with comparisons to standard Runge-Kutta methods and another structure-preserving method, demonstrate the usefulness and strengths of the methods.

关键词： Conformal symplectic Linear damping Birkhoffian method structure-preserving algorithm Dissipation preservation

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Comments on 'A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory' by Minghui Wang and Wenhao Ma

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COMPUTER PHYSICS COMMUNICATIONS 2015年 188卷 128-130页

作者： Sangwine, Stephen J. Univ Essex Sch Comp Sci & Elect Engn Colchester CO4 3SQ Essex England

Claims were made in an article by Wang and Ma in 2013 that they had devised an algorithm for the quaternion LU decomposition that was significantly faster than the LU decomposition implemented in the Quaternion Toolbox for MATLAB (QTFM). These claims have been tested and found to be unsupported by MATLAB code supplied to the author by Wang and Ma. The author's tests are presented, and test code made available as supplementary material. It is found that not only is the QTFM code faster, but that Wang and Ma's algorithm has run-time that scales with the square of the size of the matrix, whereas the algorithm in QTFM has run time approximately linear in matrix size. These findings are consistent with an inspection of the code. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Quaternion matrix LU decomposition structure-preserving algorithm

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Numerical Methods to Solve the Complex Symmetric Stabilizing Solution of the Complex Matrix Equation X+ATX-1A =Q

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数学研究 2015年第1期48卷 53-65页

作者： Yao Yao Xiao-Xia Guo School of Mathematical Sciences Ocean University of China Qingdao 266100Shandong P.R.China

When the matrices $A$ and $Q$ have special structure, the structure-preserving algorithm was used to compute the stabilizing solution of the complex matrix equation $X+A^TX^{-1}A=Q.$ In this paper, we study the numerical methods to solve the complex symmetric stabilizing solution of the general matrix equation $X+A^TX^{-1}A=Q.$ We not only establish the global convergence for the methods under an assumption, but also show the feasibility and effectiveness of them by numerical experiments.

关键词： Complex matrix complex symmetric stabilizing solution fixed-point method structure-preserving algorithm

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Continuation model predictive control on smooth manifolds

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IFAC-PapersOnLine 2015年第25期48卷 126-131页

作者： Andrew Knyazev Alexander Malyshev Mitsubishi Electric Research Labs (MERL) 201 Broadway 8th floor Cambridge MA 02139 USA

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. Continuation MPC, suggested by T. Ohtsuka in 2004, uses Krylov-Newton iterations. Continuation MPC is suitable for nonlinear problems and has been recently adopted for minimum time problems. We extend the continuation MPC approach to a case where the state is implicitly constrained to a smooth manifold. We propose an algorithm for on-line controller implementation and demonstrate its numerical effectiveness for a test problem on a hemisphere.

关键词： nonlinear model predictive control Newton-Krylov method geometric integration structure-preserving algorithm

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