A concise introduction to geometric numerical integration (Q2787626)

The rough contents of this monograph are as follows: Preface, Chapter 1: What is geometric numerical integration?, Chapter 2: Classical integrators and preservation of properties, Chapter 3: Splitting and composition methods, Chapter 4: Other types of geometric numerical integrators, Chapter 5: Long-time behavior of geometric integrators, Chapter 6: Time-splitting methods for PDEs of evolution, Bibliography and Index. The work also contains an appendix named: Some additional mathematical results, mainly devoted to Lie formalism. The MATLAB codes worked out and updated by authors are available at the website: \url{http://www.gicas.uji.es/GNIBook.html}.NEWLINENEWLINEThe authors consider in turn elementary geometric integrators, such as Euler symplectic and Störmer-Verlet algorithms, high-order structure preservation Runge-Kutta and multistep integrators, composite of basic low-order methods, splitting methods and other types of geometric integrators. Making use of backward error analysis they study some of the above mentioned methods with respect to their long time behavior. The errors propagation in long time integration is of utmost importance in many physical applications. Some initial/periodic boundary value problems attached to linear parabolic equations (mainly Schrödinger equation with different potentials) are analyzed using composite and splitting methods. Spatial discretization is carried out using finite differences or Fourier spectral collocation. The text does not contain the rigorous proofs of the most important theoretical results involved, but a huge number of well-chosen examples along with fairly profound heuristic explanations of the most relevant features of geometrical methods of numerical integration. Each chapter ends up with a rich section of exercises. Solving them is a real challenge for any reader.