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 25 articles , 1 standard article )

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  1. Castillo, Paul; Gómez, Sergio: Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations (2020)
  2. Achouri, Talha: Conservative finite difference scheme for the nonlinear fourth-order wave equation (2019)
  3. Amodio, Pierluigi; Brugnano, Luigi; Iavernaro, Felice: A note on the continuous-stage Runge-Kutta(-Nyström) formulation of Hamiltonian boundary value methods (HBVMs) (2019)
  4. Brugnano, Luigi; Gurioli, Gianmarco; Sun, Yajuan: Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (2019)
  5. Brugnano, Luigi; Iavernaro, Felice; Montijano, Juan I.; Rández, Luis: Spectrally accurate space-time solution of Hamiltonian PDEs (2019)
  6. Brugnano, Luigi; Montijano, Juan I.; Rández, Luis: On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems (2019)
  7. Jiang, Chaolong; Cai, Wenjun; Wang, Yushun: A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach (2019)
  8. Li, Haochen; Hong, Qi: An efficient energy-preserving algorithm for the Lorentz force system (2019)
  9. Li, Jiyong; Wu, Xinyuan: Energy-preserving continuous stage extended Runge-Kutta-Nyström methods for oscillatory Hamiltonian systems (2019)
  10. Li, Xiuyan; Zhang, Chiping; Ma, Qiang; Ding, Xiaohua: Arbitrary high-order EQUIP methods for stochastic canonical Hamiltonian systems (2019)
  11. Tang, Wensheng; Sun, Yajuan; Zhang, Jingjing: High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods (2019)
  12. Tang, Wensheng; Zhang, Jingjing: Symmetric integrators based on continuous-stage Runge-Kutta-Nyström methods for reversible systems (2019)
  13. Wang, Bin; Wu, Xinyuan: Exponential collocation methods for conservative or dissipative systems (2019)
  14. Wang, Bin; Wu, Xinyuan: Global error bounds of one-stage extended RKN integrators for semilinear wave equations (2019)
  15. Barletti, L.; Brugnano, L.; Frasca Caccia, G.; Iavernaro, F.: Energy-conserving methods for the nonlinear Schrödinger equation (2018)
  16. Brugnano, Luigi; Gurioli, Gianmarco; Iavernaro, Felice: Analysis of energy and quadratic invariant preserving (EQUIP) methods (2018)
  17. Brugnano, Luigi; Gurioli, Gianmarco; Iavernaro, Felice; Weinmüller, Ewa B.: Line integral solution of Hamiltonian systems with holonomic constraints (2018)
  18. Cai, Wenjun; Li, Haochen; Wang, Yushun: Partitioned averaged vector field methods (2018)
  19. Liu, Changying; Wu, Xinyuan; Shi, Wei: New energy-preserving algorithms for nonlinear Hamiltonian wave equation equipped with Neumann boundary conditions (2018)
  20. Wang, Bin; Meng, Fanwei; Fang, Yonglei: Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations (2017)

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