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Recent developments in structure-preserving algorithms for oscillatory differential equations

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The main theme of this book is recent progress in structure-preserving algorithms for solving initial value problems of oscillatory differential equations arising in a variety of research areas, such as astronomy, theoretical physics, electronics, quantum mechanics and engineering. It systematically describes the latest advances in the development of structure-preserving integrators for oscillatory differential  Read more...

Abstract: The main theme of this book is recent progress in structure-preserving algorithms for solving initial value problems of oscillatory differential equations arising in a variety of research areas, such as astronomy, theoretical physics, electronics, quantum mechanics and engineering. It systematically describes the latest advances in the development of structure-preserving integrators for oscillatory differential equations, such as structure-preserving exponential integrators, functionally fitted energy-preserving integrators, exponential Fourier collocation methods, trigonometric collocation methods, and symmetric and arbitrarily high-order time-stepping methods. Most of the material presented here is drawn from the recent literature. Theoretical analysis of the newly developed schemes shows their advantages in the context of structure preservation. All the new methods introduced in this book are proven to be highly effective compared with the well-known codes in the scientific literature. This book also addresses challenging problems at the forefront of modern numerical analysis and presents a wide range of modern tools and techniques

Author(s): Wu K., Wang BIN
Publisher: SPRINGER Verlag, SINGAPOR
Year: 2018

Language: English
Pages: 356
Tags: Mathematics.;Algorithms.;Computational complexity.;Mathematics of Algorithmic Complexity;Complexity;Mathematics of Algorithmic Complexity.;Complexity.

Content: Intro
Preface
Contents
1 Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems
1.1 Introduction
1.2 Functionally-Fitted Continuous Finite Element Methods …
1.3 Interpretation as Continuous-Stage Runge-Kutta …
1.4 Implementation Issues
1.5 Numerical Experiments
1.6 Conclusions and Discussions
References
2 Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems
2.1 Introduction
2.2 Discrete Gradient Integrators
2.3 Exponential Discrete Gradient Integrators
2.4 Symmetry and Convergence of the EAVF Integrator. 2.5 Problems Suitable for EAVF2.5.1 Highly Oscillatory Nonseparable Hamiltonian Systems
2.5.2 Second-Order (Damped) Highly Oscillatory System
2.5.3 Semi-discrete Conservative or Dissipative PDEs
2.6 Numerical Experiments
2.7 Conclusions and Discussions
References
3 Exponential Fourier Collocation Methods for First-Order Differential Equations
3.1 Introduction
3.2 Formulation of EFCMs
3.2.1 Local Fourier Expansion
3.2.2 Discretisation
3.2.3 The Exponential Fourier Collocation Methods
3.3 Connections with Some Existing Methods
3.3.1 Connections with HBVMs and Gauss Methods. 3.3.2 Connection Between EFCMs and Radau IIA Methods3.3.3 Connection Between EFCMs and TFCMs
3.4 Properties of EFCMs
3.4.1 The Hamiltonian Case
3.4.2 The Quadratic Invariants
3.4.3 Algebraic Order
3.4.4 Convergence Condition of the Fixed-Point Iteration
3.5 A Practical EFCM and Numerical Experiments
3.6 Conclusions and Discussions
References
4 Symplectic Exponential Runge-Kutta Methods for Solving Nonlinear Hamiltonian Systems
4.1 Introduction
4.2 Symplectic Conditions for ERK Methods
4.3 Symplectic ERK Methods
4.4 Numerical Experiments
4.5 Conclusions and Discussions.