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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].

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
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