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ORDER PROPERTIES AND CONSTRUCTION OF symplectic runge-kutta methods

Journal of Computational Mathematics 2000年第6期18卷 645-656页

作者： Shou-fu Li (Institute for Computational and Applied Mathematics, Xiangtan University, Xiangtan 411105, China) 湘潭大学湖南湘潭 411105

The main results of this paper are as follows: (1) Suppose an s stage runge-kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p + l(1 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 0, 1, 2. (2) Suppose an s stage runge-kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 s. Then this method is symplectic if and only if either p = s or the nonlinear stability matrix M of the method has an (s - p) x (s - p) chief submatrix (M) over cap = 0. (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic runge-kutta methods, and a software has been designed, by means of which, the coefficients of s stage symplectic runge-kutta methods satisfying C(p),D(p) and B(2p + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.

关键词： numerical analysis symplectic runge-kutta methods simplifying conditions order results

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symplectic runge-kutta methods for The Linear Quadratic Regulator Problem

Symplectic Runge-Kutta Methods for The Linear Quadratic Regu...

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保结构算法2004

作者： Guang-Da Hu Department of Control Science and Engineering P. O. Box 329, Harbin Institute of Technology Harbin, 150001. China

＜正＞In this paper, numerical methods for the linear quadratic regulator （LQR） problem are investigated from the viewpoint of geometry. The differential matrix Riccati equation for the linear quadratic regulator problem is transformed into a linear differential Hamiltonian system. We show that the linear differential Hamiltonian system with two different terminal conditions are on symplectic group. The two different terminal conditions relate to two different performance indexes for the same regulator problem. Then symplectic runge-kutta methods can be applied to the linear differential Hamiltonian system, which keep the numerical solution on the symplectic group. Numerical examples are given to illustrate the performance of the numerical methods.

关键词： optimal control Hamiltonian system. Riccati equation symplectic group symplectic runge-kutta methods

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Explicit Multi-symplectic methods for Hamiltonian Wave Equations

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Communications in Computational Physics 2007年第4期2卷 662-683页

作者： Jialin Hong Shanshan Jiang Chun Li Hongyu Liu State Key Laboratory of Scientific and Engineering Computing Institute of Computational MathematicsAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100080P.R.China Graduate School Chinese Academy of SciencesBeijing 100080P.R.China Department of Mathematics The Chinese University of Hong KongShatinN.T.Hong Kong.

In this paper,based on the multi-symplecticity of concatenating symplectic runge-kutta-Nystrom(SRKN)methods and symplectic runge-kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model *** comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.

关键词： Hamiltonian wave equations multi-symplectic integration symplectic runge-kutta methods symplectic runge-kutta-Nystrom methods.

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Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

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Advances in Applied Mathematics and Mechanics 2023年第5期15卷 1233-1255页

作者： Yuezheng Gong Qi Hong Chunwu Wang Yushun Wang School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjingJiangsu 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjingJiangsu 211106China Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems School of Mathematical SciencesNanjing Normal UniversityNanjingJiangsu 210023China

In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm *** QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial *** the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic runge-kutta methods in time,which arrives at a class of fully discrete *** the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and *** numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed *** presented numerical strategy makes it possible to directly apply a special class of runge-kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

关键词： Camassa-Holm equation quadratic auxiliary variable high-order energy-preserving schemes symplectic runge-kutta methods

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