出版社:科学出版社
出版日期:2013-4
ISBN:9787030355201
页数:236页
章节摘录
Chapter 1 RungeKutta (Nystr.m) Methods for Oscillatory Differential EquationsIn this chapter we .rst survey Runge Kutta (RK) methods for initial value prob-lems of .rst-order ordinary differential equations. For the purpose of deriving order conditions, the rooted tree theory is set up. For second-order differential equations, Runge Kutta Nystr.m (RKN) methods are formulated, and their order conditions are obtained based on the Nystr.m tree theory. For oscillatory differential equations, the dispersion and dissipation of classical numerical methods are examined. We also recall the symplectic RK and RKN methods for Hamiltonian systems. Finally, we make some comments on structure-preserving methods for solving oscillatory prob-lems.1.1 RK Methods, Rooted Trees, B-Series and Order ConditionsWe start with an initial value problem of ordinary differential equations de.ned on the interval [x0,xend]:y=f(x,y),y(x0)=y0,(1.1)where y∈Rdand f:R×Rd→Rd. From the existence theory of ordinary differ-ential equations, the problem (1.1) has a unique solution on [x0,xend]if the function f(x,y)is continuous in its .rst variable and satis.es a Lipschitz condition in its second variable (see Butcher [3]). However, on most occasions, the true solution to the initial value problem (1.1) arising in applications, is not accessible even though it exists. Therefore it becomes common practice to solve the initial value problem(1.1) by numerical approaches, among which the classical RK methods are most popular.RK methods were developed by Runge [17], Heun [12] and Kutta [14]. Although a number of different approaches have been employed in the analysis of RK meth-ods, the one used in this chapter is that established by Butcher [1, 2], following on from the work of Gill [5] and Merson [15].
内容概要
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.
书籍目录
1 Runge-Kutta(-Nyström)Methods for Oscillatory Differential Equations1.1 RK Methods,Rooted Trees,B-Series and Order Conditions1.2 RKN Methods,Nyström Trees and Order Conditions1.2.1 Formulation of the Scheme1.2.2 NyströmTrees andOrderConditions1.2.3 The Special Case in Absence of the Derivative1.3 Dispersion and Dissipation of RK(N)Methods1.3.1 RK Methods1.3.2 RKN Methods1.4 Symplectic Methods for Hamiltonian Systems1.5 Comments on Structure-Preserving Algorithms for Oscillatory ProblemsReferences2 ARKN Methods2.1 Traditional ARKN Methods2.1.1 Formulation of the Scheme2.1.2 OrderConditions2.2 Symplectic ARKN Methods2.2.1 SymplecticityConditions forARKNIntegrators2.2.2 Existence of Symplectic ARKN Integrators2.2.3 Phase and Stability Properties of Method SARKN1s22.2.4 Nonexistence of Symmetric ARKN Methods2.2.5 Numerical Experiments2.3 Multidimensional ARKN Methods2.3.1 Formulation of the Scheme2.3.2 OrderConditions2.3.3 Practical Multidimensional ARKN MethodsReferences3 ERKN Methods3.1 ERKN Methods3.1.1 Formulation of Multidimensional ERKN Methods3.1.2 Special Extended Nyström Tree Theory3.1.3 OrderConditions3.2 EFRKN Methods and ERKN Methods3.2.1 One-Dimensional Case3.2.2 Multidimensional Case3.3 ERKN Methods for Second-Order Systems with Variable Principal Frequency Matrix3.3.1 Analysis Through an Equivalent System3.3.2 Towards ERKN Methods3.3.3 Numerical IllustrationsReferences4 Symplectic and Symmetric Multidimensional ERKN Methods4.1 Symplecticity and Symmetry Conditions for Multidimensional ERKNIntegrators4.1.1 SymmetryConditions4.1.2 SymplecticityConditions4.2 ConstructionofExplicitSSMERKNIntegrators4.2.1 Two Two-Stage SSMERKN Integrators of Order Two4.2.2 AThree-StageSSMERKNIntegratorofOrderFour4.2.3 Stability and Phase Properties of SSMERKN Integrators4.3 Numerical Experiments4.4 ERKN Methods for Long-Term Integration of Orbital Problems4.5 Symplectic ERKN Methods for Time-Dependent Second-Order Systems4.5.1 Equivalent Extended Autonomous Systems for Nonautonomous Systems4.5.2 Symplectic ERKN Methods for Time-Dependent HamiltonianSystems4.6 Concluding RemarksReferences5 Two-Step Multidimensional ERKN Methods5.1 The Scheifele Two-Step Methods5.2 Formulation of TSERKN Methods5.3 OrderConditions5.3.1 B-Series onSENT5.3.2 One-StepFormulation5.3.3 OrderConditions5.4 Construction of Explicit TSERKN Methods5.4.1 A Method with Two Function Evaluations per Step5.4.2 Methods with Three Function Evaluations per Step5.5 Stability and Phase Properties of the TSERKN Methods5.6 Numerical ExperimentsReferences6 Adapted Falkner-Type Methods6.1 Falkner's Methods6.2 Formulation of the Adapted Falkner-Type Methods6.3 ErrorAnalysis6.4 Stability6.5 Numerical ExperimentsAppendix A Derivation of Generating Functions(6.14)and(6.15)Appendix B Proof of(6.24)References7 Energy-Preserving ERKN Methods7.1 The Average-Vector-Field Method7.2 Energy-Preserving ERKN Methods7.2.1 Formulation of the AAVF methods7.2.2 A Highly Accurate Energy-Preserving Integrator7.2.3 Two Properties of the Integrator AAVF-GL7.3 Numerical Experiment on the Fermi-Pasta-Ulam ProblemReferences8 Effective Methods for Highly Oscillatory Second-Order Nonlinear Differential Equations8.1 Numerical Consideration of Highly Oscillatory Second-Order DifferentialEquations8.2 The Asymptotic Method for Linear Systems8.3 Waveform Relaxation(WR)Methods for Nonlinear SystemsReferences9 Extended Leap-Frog Methods for HamiltonianWave Equations9.1 Conservation Laws and Multi-Symplectic Structures of Wave Equations9.1.1 Multi-Symplectic Conservation Laws9.1.2 ConservationLaws forWaveEquations9.2 ERKNDiscretizationofWaveEquations9.2.1 Multi-Symplectic Integrators9.2.2 Multi-Symplectic Extended RKN Discretization9.3 Explicit Extended Leap-Frog Methods9.3.1 Eleap-Frog I:An Explicit Multi-Symplectic ERKN Scheme9.3.2 Eleap-Frog II:An Explicit Multi-Symplectic ERKN-PRK Scheme9.3.3 Analysis of Linear Stability9.4 Numerical Experiments9.4.1 TheConservationLaws andtheSolution9.4.2 DispersionAnalysisReferencesAppendix First and Second Symposiums on Structure-Preserving Algorithms for Differential Equations,August 2011,June 2012,NanjingIndex
编辑推荐
Xinyuan Wu、Xiong You、Bin Wang所著的《振荡微分方程的保结构算法(英文版)(精)》反映了二阶振荡微分方程保结构数值解法研究的最近进展和发展动向，系统阐述了作者及其合作者近五年在常微分方程的ARKN方法、ERKN方法、两步ERKN方法、Falkner型方法、辛方法、对称方法、保能量方法以及偏微分方程多辛方法等方面的重要研究成果。从经典的普适性方法到面向于振荡问题的拟合型方法；从单步法到多步法；从常微分方程的数值解法到偏微分方程的多辛算法。
作者简介
《振荡微分方程的保结构算法(英文版)》内容简介：Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for secondorder 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.