LIMbook

Line Integral Methods for Conservative Problems. The book explains the numerical solution of differential equations within the framework of geometric integration, a branch of numerical analysis that devises numerical methods able to reproduce (in the discrete solution) relevant geometric properties of the continuous vector field. The book focuses on a large set of differential systems named conservative problems, particularly Hamiltonian systems. Assuming only basic knowledge of numerical quadrature and Runge-Kutta methods, this self-contained book begins with an introduction to the line integral methods. It describes numerous Hamiltonian problems encountered in a variety of applications and presents theoretical results concerning the main instance of line integral methods: the energy-conserving Runge-Kutta methods, also known as Hamiltonian boundary value methods (HBVMs). The authors go on to address the implementation of HBVMs in order to recover in the numerical solution what was expected from the theory. The book also covers the application of HBVMs to handle the numerical solution of Hamiltonian partial differential equations (PDEs) and explores extensions of the energy-conserving methods. With many examples of applications, this book provides an accessible guide to the subject yet gives you enough details to allow concrete use of the methods. MATLAB codes for implementing the methods are available online


References in zbMATH (referenced in 59 articles , 1 standard article )

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  1. Bo, Yonghui; Cai, Jiaxiang; Cai, Wenjun; Wang, Yushun: The exponential invariant energy quadratization approach for general multi-symplectic Hamiltonian PDEs (2022)
  2. De Marinis, Arturo; Iavernaro, Felice; Mazzia, Francesca: A minimum-time obstacle-avoidance path planning algorithm for unmanned aerial vehicles (2022)
  3. Hu, Dongdong; Cai, Wenjun; Gu, Xian-Ming; Wang, Yushun: Efficient energy preserving Galerkin-Legendre spectral methods for fractional nonlinear Schrödinger equation with wave operator (2022)
  4. Jiang, Chaolong; Cui, Jin; Qian, Xu; Song, Songhe: High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation (2022)
  5. Jiang, Chaolong; Qian, Xu; Song, Songhe; Cui, Jin: Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation (2022)
  6. Mei, Lijie; Huang, Li; Wu, Xinyuan: Energy-preserving continuous-stage exponential Runge-Kutta integrators for efficiently solving Hamiltonian systems (2022)
  7. Shi, Wei; Liu, Kai: A dissipation-preserving integrator for damped oscillatory Hamiltonian systems (2022)
  8. Akrivis, Georgios; Li, Dongfang: Structure-preserving Gauss methods for the nonlinear Schrödinger equation (2021)
  9. Frasca-Caccia, Gianluca; Hydon, Peter E.: Numerical preservation of multiple local conservation laws (2021)
  10. Fu, Ting; Zhang, Mingqian; Liu, Kai: An integral evolution formula of boundary value problem for wave equations (2021)
  11. Hu, Dongdong; Cai, Wenjun; Wang, Yushun: Two linearly implicit energy preserving exponential scalar auxiliary variable approaches for multi-dimensional fractional nonlinear Schrödinger equations (2021)
  12. Hu, Dongdong; Gong, Yuezheng; Wang, Yushun: On convergence of a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrödinger equation and its fast implementation (2021)
  13. Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Ya. D.: Computation of higher order Lie derivatives on the infinity computer (2021)
  14. Jiang, Chaolong; Wang, Yushun; Gong, Yuezheng: Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations (2021)
  15. Liu, Kai; Fu, Ting; Shi, Wei: A dissipation-preserving scheme for damped oscillatory Hamiltonian systems based on splitting (2021)
  16. Xie, Jianqiang; Wang, Quanxiang; Zhang, Zhiyue: Linear implicit finite difference methods with energy conservation property for space fractional Klein-Gordon-Zakharov system (2021)
  17. Amodio, Pierluigi; Brugnano, Luigi; Iavernaro, Felice: Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE problems (2020)
  18. Barletti, Luigi; Brugnano, Luigi; Tang, Yifa; Zhu, Beibei: Spectrally accurate space-time solution of Manakov systems (2020)
  19. Brugnano, Luigi; Iavernaro, Felice; Zhang, Ruili: Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles (2020)
  20. Castillo, Paul; Gómez, Sergio: Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations (2020)

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