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 10 articles , 1 standard article )

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  1. Brugnano, Luigi; Gurioli, Gianmarco; Sun, Yajuan: Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (2019)
  2. Brugnano, Luigi; Gurioli, Gianmarco; Iavernaro, Felice: Analysis of energy and quadratic invariant preserving (EQUIP) methods (2018)
  3. Brugnano, Luigi; Gurioli, Gianmarco; Iavernaro, Felice; Weinmüller, Ewa B.: Line integral solution of Hamiltonian systems with holonomic constraints (2018)
  4. Cai, Wenjun; Li, Haochen; Wang, Yushun: Partitioned averaged vector field methods (2018)
  5. Wang, Bin; Meng, Fanwei; Fang, Yonglei: Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations (2017)
  6. Wang, Bin; Wu, Xinyuan; Meng, Fanwei: Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations (2017)
  7. Yan, Jin-Liang; Zhang, Qian; Zhang, Zhi-Yue; Liang, Dong: A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation (2017)
  8. Brugnano, Luigi; Iavernaro, Felice: Line integral methods for conservative problems (2016)
  9. Li, Haochen; Wang, Yushun: A discrete line integral method of order two for the Lorentz force system (2016)
  10. Zhang, Wei; Simos, T. E.: A high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation (2016)