Budd, C. J.; Piggott, M. D. Geometric integration and its applications. (English) Zbl 1062.65134 Cucker, F. (ed.), Handbook of numerical analysis. Vol. XI. Special volume: Foundations of computational mathematics. Amsterdam: North-Holland (ISBN 0-444-51247-0/hbk). Handb. Numer. Anal. 11, 35-139 (2003). The author develop the theory of geometric integration methods. They prove the importance of the qualitative features of solutions in the construction of numerical methods. Euler, Runge-Kutta, and discretisation (finite-differences) methods are studied under the aspect to preserve the properties of symplecticity, symmetry, singularity and others. The discrete form of Noether’s theorems ensures the conservation of invariance of Lagrangian transformation groups. For the nonlinear Schrödinger equation and the nonlinear diffusion equation the aspect of geometric integration is studied. This article presents a synthesis of many recent works which report the aspect of geometrical integration.For the entire collection see [Zbl 1044.65002]. Reviewer: Ivan Secrieru (Chişinău) Cited in 33 Documents MSC: 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 34A26 Geometric methods in ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 35K55 Nonlinear parabolic equations 34A34 Nonlinear ordinary differential equations and systems Keywords:symplectic methods; Hamiltonian ordinary and PDE; nonlinear diffusion equation; geometric integration; nonlinear Schrödinger equation; Euler method; Runge-Kutte method; finite difference method PDFBibTeX XMLCite \textit{C. J. Budd} and \textit{M. D. Piggott}, Handb. Numer. Anal. 11, 35--139 (2003; Zbl 1062.65134)