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Structure-Preserving Algorithms for Oscillatory Differential Equations

verfasst von: Xinyuan Wu, Xiong You, Bin Wang

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations by using theoretical analysis and numerical validation. Structure-preserving algorithms for differential equations, especially for oscillatory differential equations, play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering. The book discusses novel advances in the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving methods, etc. for oscillatory differential equations.

The work is intended for scientists, engineers, teachers and students who are interested in structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing University; Xiong You is an associate professor at Nanjing Agricultural University; Bin Wang is a joint Ph.D student of Nanjing University and University of Cambridge.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Runge–Kutta (–Nyström) Methods for Oscillatory Differential Equations

Abstract

Chapter 1 is a brief introduction to classical approach to the numerical treatment of initial value problems (IVPs) of ordinary differential equations. For first-order differential equations, Runge–Kutta (RK) methods are presented. For the purpose of deriving order conditions, the rooted tree theory is set up. For second-order differential equations, Runge–Kutta–Nyström (RKN) methods are formulated, and their order conditions are obtained based on the Nyström tree theory. For oscillatory differential equations, the dispersion and dissipation of classical numerical methods are examined. Symplectic RK and RKN methods for Hamiltonian systems are also recalled. Some comments are given on structure-preserving methods, especially exponentially/trigonometrically fitted methods, for solving oscillatory problems. Finally, the contents and the structure of the whole book are described.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 2. ARKN Methods

Abstract

Chapter 2 investigates the adapted Runge–Kutta–Nyström (ARKN) methods proposed by Franco (2002) for the system of oscillatory second-order differential equations y″+ω ² y=f(y,y′), where ω>0 is the main frequency. Based on the internal stages of the traditional RKN methods, ARKN methods adopt a new form of updates which incorporate the special oscillatory structure of the system. Order conditions for ARKN methods are derived by means of the Nyström tree theory. The symplecticity conditions for ARKN methods are obtained. It is also shown that an ARKN method cannot be symmetric. The effectiveness of a one-stage symplectic ARKN method is illustrated by Duffing equations, the Fermi–Pasta–Ulam problem and the “almost periodic” orbit problem. On the basis of the matrix-variation-of-constants formula established by Wu et al. (2009), multidimensional ARKN methods are developed for the more general oscillatory system y″+My=f(y,y′) with a positive semi-definite (not necessarily symmetric) main frequency matrix M. These methods do not rely on the decomposition of M so that they are applicable to oscillatory systems with a positive semi-definite (but not symmetric) frequency matrix.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 3. ERKN Methods

Abstract

Chapter 3 proposes and investigates extended Runge–Kutta–Nyström (ERKN) methods for the oscillatory second-order system y″+My=f(y), where the frequency matrix M is positive semi-definite. ERKN methods make full use of the special structure of the equation introduced by the linear term My so that they are exact in both the internal stages and the updates when applied to the multidimensional homogeneous system y″+My=0. In order to derive the order conditions for ERKN methods, a special extended Nyström tree (SEN-tree) theory and the related B-series theory are developed. Then the multidimensional exponential-fitting methods are presented and the relation between ERKN methods and exponentially fitted methods is discussed. Finally, ERKN methods for systems with variable principal frequency matrices are investigated and are illustrated by numerical examples including the time-dependent anharmonic undamped oscillator.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 4. Symplectic and Symmetric Multidimensional ERKN Methods

Abstract

Symplectic and symmetric Runge–Kutta–Nyström-type methods have been shown to have excellent behavior in the long-term integration of Hamiltonian systems. Chapter 4 focuses on the investigation of symplectic and symmetric multi-frequency and multidimensional extended Runge–Kutta–Nyström (SSMERKN) integrators. The symplecticity and symmetry conditions for multidimensional ERKN methods are obtained. When the principal frequency matrix vanishes, they reduce to those for the traditional RKN methods with constant coefficients. Some practical SSMERKN integrators are derived. The stability and phase properties of SSMERKN integrators are analyzed. A technique is developed for transforming a non-autonomous Hamiltonian system into an equivalent autonomous Hamiltonian system in an extended phase space. Symplectic multidimensional ERKN methods applied to the equivalent system are shown to preserve the extended energy very well. Numerical experiments are carried out on three nonlinear wave equations and the Fermi–Pasta–Ulam problem.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 5. Two-Step Multidimensional ERKN Methods

Abstract

In Chap. 5, multidimensional two-step extended Runge–Kutta–Nyström-type (TSERKN) methods are developed for solving the oscillatory second-order system y″+My=f(x,y), where M∈ℝ^d×d is a symmetric positive semi-definite matrix that implicitly contains the frequencies of the problem. The new methods inherit the framework of two-step hybrid methods and are adapted to the special features of the true flows in both the internal stages and the updates. Based on the SEN-tree theory in Chap. 3, order conditions for the TSERKN methods are derived via the B-series defined on the set SENT of trees and the B ^f-series defined on the subset SENT ^f of SENT. Three explicit TSERKN methods are constructed and their stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of the new methods in comparison with the well-known high quality methods proposed in the literature.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 6. Adapted Falkner-Type Methods

Abstract

Chapter 6 establishes the multidimensional adapted Falkner-type methods for the oscillatory second-order system y″+My=f(x,y) with a symmetric positive semi-definite principal frequency matrix M∈ℝ^d×d. Adapted generating functions are formulated for deriving the coefficients of adapted Falkner-type methods. Based on the discrete Gronwall’s inequality, uniform bounds for the local truncation errors of the solution and the derivative are obtained, respectively. These error bounds turn out to be independent of the frequency matrix M. Zero-stability as well as linear stability of adapted Falkner-type methods is also analyzed. The high efficiency of adapted Falkner-type methods is illustrated by numerical examples such as the coupled oscillators, the sine-Gordon equation and a nonlinear wave equation.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 7. Energy-Preserving ERKN Methods

Abstract

Chapter 7 is concerned with the energy-preserving numerical integration for the system of oscillatory second-order differential equations \(\ddot{q}+Mq=f(q)\), where M is a symmetric positive semi-definite matrix and f(q)=−∇U(q). Based on the traditional average-vector-field (AVF) methods, adapted average-vector-field (AAVF) methods are developed. A discretization with a quadrature formula leads to a highly accurate energy-preserving ERKN-type AAVF integrator. This integrator is symmetric and is shown to preserve the Hamiltonian H if U(q) is a polynomial of degree s≤6. In the long-term integration of the well-known Fermi–Pasta–Ulam problem, the integrator is shown to preserve the energy more accurately than some existing methods in the literature. Resonance instabilities and energy exchange between stiff components are also illustrated.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 8. Effective Methods for Highly Oscillatory Second-Order Nonlinear Differential Equations

Abstract

Numerical treatment of highly oscillatory problems has received an increasing attention in recent years. Chapter 8 is devoted to effective integration of highly oscillatory second-order differential equations. Based on the matrix-variation-of-constants formula, the adapted asymptotic method for the highly oscillatory linear system \(\ddot{q}+Mq=g(t)\) is developed. Local truncation errors are estimated. The long-term behavior of the adapted asymptotic method is illustrated numerically for the frequency as high as 10⁸. Then the approach is extended to the nonlinear system \(\ddot{q}+Mq=f(t,q, \dot{q})\) with the aid of the waveform relaxation procedure. When applied to Duffing’s equation for different frequencies, this waveform relaxation-asymptotic method (WRAM) is shown to be very effective. The higher the frequency is, the more accurate it is.

Xinyuan Wu, Xiong You, Bin Wang

Chapter 9. Extended Leap-Frog Methods for Hamiltonian Wave Equations

Abstract

Structure-preserving algorithms, or multi-symplectic methods for partial differential equations, though less developed, have been considered as important as those for ordinary differential equations. In Chap. 9, the idea of ERKN methods for oscillatory ordinary differential equations is extended to the integration of oscillatory partial differential equations. Multi-symplectic discretizations of the Hamiltonian wave equations and the corresponding discrete conservation laws are investigated. The discretization by two symplectic ERKN methods in time and space, or by a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method in space, leads to a multi-symplectic integrator. Two explicit multi-symplectic extended leap-frog integrators are derived. The numerical stability and dispersive properties of the integrators are analyzed. The two integrators are applied to the linear wave equation and the sine-Gordon equation.

Xinyuan Wu, Xiong You, Bin Wang

Structure-Preserving Algorithms for Oscillatory Differential Equations

Xinyuan Wu, Xiong You, Bin Wang

Backmatter

Titel: Structure-Preserving Algorithms for Oscillatory Differential Equations
verfasst von: Xinyuan Wu
Xiong You
Bin Wang
Copyright-Jahr: 2013
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-35338-3
Print ISBN: 978-3-642-35337-6
DOI: https://doi.org/10.1007/978-3-642-35338-3

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