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Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

  • Book
  • © 2021

Overview

Authors:
  1. Xinyuan Wu

    1. Department of Mathematics, Nanjing University, Nanjing, China

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    You can also search for this author in PubMed   Google Scholar

  2. Bin Wang

    1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Establishes structure-preserving algorithms for differential equations

  • Presents theoretical derivations and mathematical analysis

  • Provides high-performance numerical simulations

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. Oscillation-Preserving Integrators for Highly Oscillatory Systems of Second-Order ODEs

    • Xinyuan Wu, Bin Wang
    Pages 1-45

  3. Continuous-Stage ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions

    • Xinyuan Wu, Bin Wang
    Pages 47-74

  4. Stability and Convergence Analysis of ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions

    • Xinyuan Wu, Bin Wang
    Pages 75-122

  5. Functionally-Fitted Energy-Preserving Integrators for Poisson Systems

    • Xinyuan Wu, Bin Wang
    Pages 123-146

  6. Exponential Collocation Methods for Conservative or Dissipative Systems

    • Xinyuan Wu, Bin Wang
    Pages 147-177

  7. Volume-Preserving Exponential Integrators

    • Xinyuan Wu, Bin Wang
    Pages 179-211

  8. Global Error Bounds of One-Stage Explicit ERKN Integrators for SemilinearWave Equations

    • Xinyuan Wu, Bin Wang
    Pages 213-233

  9. Linearly-Fitted Conservative (Dissipative) Schemes for Nonlinear Wave Equations

    • Xinyuan Wu, Bin Wang
    Pages 235-261

  10. Energy-Preserving Schemes for High-Dimensional Nonlinear KG Equations

    • Xinyuan Wu, Bin Wang
    Pages 263-297

  11. High-Order Symmetric Hermite–Birkhoff Time Integrators for Semilinear KG Equations

    • Xinyuan Wu, Bin Wang
    Pages 299-349

  12. Symplectic Approximations for Efficiently Solving Semilinear KG Equations

    • Xinyuan Wu, Bin Wang
    Pages 351-391

  13. Continuous-Stage Leap-Frog Schemes for Semilinear Hamiltonian Wave Equations

    • Xinyuan Wu, Bin Wang
    Pages 393-426

  14. Semi-Analytical ERKN Integrators for Solving High-Dimensional Nonlinear Wave Equations

    • Xinyuan Wu, Bin Wang
    Pages 427-458

  15. Long-Time Momentum and Actions Behaviour of Energy-Preserving Methods for Wave Equations

    • Xinyuan Wu, Bin Wang
    Pages 459-496

  16. Back Matter

    Pages 497-499
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Keywords

About this book

The idea of structure-preserving algorithms appeared in the 1980's. The new paradigm brought many innovative changes. The new paradigm wanted to identify the long-time behaviour of the solutions or the existence of conservation laws or some other qualitative feature of the dynamics. Another area that has kept growing in importance within Geometric Numerical Integration is the study of highly-oscillatory problems: problems where the solutions are periodic or quasiperiodic and have to be studied in time intervals that include an extremely large number of periods. As is known, these equations cannot be solved efficiently using conventional methods. A further study of novel geometric integrators has become increasingly important in recent years. The objective of this monograph is to explore further geometric integrators for highly oscillatory problems that can be formulated as systems of ordinary and partial differential equations.

Facing challenging scientific computational problems, this book presents some new perspectives of the subject matter based on theoretical derivations and mathematical analysis, and provides high-performance numerical simulations. In order to show the long-time numerical behaviour of the simulation, all the integrators presented in this monograph have been tested and verified on highly oscillatory systems from a wide range of applications in the field of science and engineering. They are more efficient than existing schemes in the literature for differential equations that have highly oscillatory solutions. 



This book is useful to researchers, teachers, students and engineers who are interested in Geometric Integrators and their long-time behaviour analysis for differential equations with highly oscillatory solutions. 

Reviews

“This is an advanced and specialized text. … Each part has discussions of numerical experiments and examples. The numerical examples show the preliminaries for computations, describe the method, and then present numerical results. … many references are provided.” (Bill Satzer, MAA Reviews, January 5, 2023)

Authors and Affiliations

  • Department of Mathematics, Nanjing University, Nanjing, China

    Xinyuan Wu

  • School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

    Bin Wang

About the authors

Xinyuan Wu, a Professor in Department of Mathematics, Nanjing University. His research interests focus on geometric algorithms for differential equations, numerical methods for stiff problems and numerical methods for algebraic systems. ​In 2017, Wu was awarded with the highest distinction of “Honorary Fellowship” from European Society of Computational Methods in Science and Engineering for the outstanding contribution in the fields of Numerical Analysis and Applied Mathematics. Wu attended the school of Mathematics at the University of Tübingen for study and research from Janurary 19th 2002 to Janurary 20th 2003.



Bin Wang, a Professor in Department of Mathematics and Statistics, Xi'an Jiaotong University. His research interests focus on various structure-preserving algorithms as well as numerical methods for differential equation, especially the numerical computation and analysis of Hamilton ordinary differential equation and partial differential equation. Wang was awarded by Alexander von Humboldt Foundation (2017–2019).

Bibliographic Information

  • Book Title: Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

  • Authors: Xinyuan Wu, Bin Wang

  • DOI: https://doi.org/10.1007/978-981-16-0147-7

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021

  • Hardcover ISBN: 978-981-16-0146-0Published: 29 September 2021

  • Softcover ISBN: 978-981-16-0149-1Published: 30 September 2022

  • eBook ISBN: 978-981-16-0147-7Published: 28 September 2021

  • Edition Number: 1

  • Number of Pages: XVIII, 499

  • Number of Illustrations: 103 b/w illustrations, 83 illustrations in colour

  • Additional Information: Jointly published with Science Press Beijing, China

  • Topics: Analysis, Numerical Analysis, Dynamical Systems and Ergodic Theory, Vibration, Dynamical Systems, Control

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