Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

1. Front Matter

   Pages i-xv

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2. Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems

   • Xinyuan Wu, Bin Wang

   Pages 1-28

3. Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems

   • Xinyuan Wu, Bin Wang

   Pages 29-53

4. Exponential Fourier Collocation Methods for First-Order Differential Equations

   • Xinyuan Wu, Bin Wang

   Pages 55-84

5. Symplectic Exponential Runge–Kutta Methods for Solving Nonlinear Hamiltonian Systems

   • Xinyuan Wu, Bin Wang

   Pages 85-106

6. High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems

   • Xinyuan Wu, Bin Wang

   Pages 107-133

7. The Construction of Arbitrary Order ERKN Integrators via Group Theory

   • Xinyuan Wu, Bin Wang

   Pages 135-165

8. Trigonometric Collocation Methods for Multi-frequency and Multidimensional Oscillatory Systems

   • Xinyuan Wu, Bin Wang

   Pages 167-192

9. A Compact Tri-Colored Tree Theory for General ERKN Methods

   • Xinyuan Wu, Bin Wang

   Pages 193-220

10. An Integral Formula Adapted to Different Boundary Conditions for Arbitrarily High-Dimensional Nonlinear Klein–Gordon Equations

    • Xinyuan Wu, Bin Wang

    Pages 221-250

11. An Energy-Preserving and Symmetric Scheme for Nonlinear Hamiltonian Wave Equations

    • Xinyuan Wu, Bin Wang

    Pages 251-268

12. Arbitrarily High-Order Time-Stepping Schemes for Nonlinear Klein–Gordon Equations

    • Xinyuan Wu, Bin Wang

    Pages 269-316

13. An Essential Extension of the Finite-Energy Condition for ERKN Integrators Solving Nonlinear Wave Equations

    • Xinyuan Wu, Bin Wang

    Pages 317-342

14. Back Matter

    Pages 343-345

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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 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.

• structure-preserving exponential integrators
• functionally-fitted energy-preserving integrators
• exponential Fourier collocation methods
• trigonometric collocation methods
• symmetric and arbitrarily high-order time-stepping methods
• algorithmic complexity
• complexity

• Department of Mathematics, Nanjing University, Nanjing, China

  Xinyuan Wu

• School of Mathematical Sciences, Qufu Normal University, Qufu, China

  Bin Wang

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