本帖最后由 VibInfo 于 2016-4-14 15:38 编辑
Geometric integration is the discipline concerned with numerical discretization methods that respect qualitative features of the underlying differential equations.In a sense, geometric integration attempts to unify two worlds: the qualitative world of pure mathematics, with an emphasis on geometry, and the quantitative world of numerical analysis. Typical themes in geometric integration are
Symplectic methods for Hamiltonian ordinary differential equations;
Multisymplectic methods and Moser-Veselov integrators for Hamiltonian partial differential equations;
Lie-group methods, for differential equations evolving in Lie groups or on homogeneous manifolds;
Methods for differential equations which conserve energy, volume, angular momentum or other invariants;
Methods that respect laws of physics and the Stokes theorem by employing finiteelement discretization on topological complexes.
Although geometric integration evolves very fast, there exist a number of excellent references which are a good place to start if you wish to learn more:
1. C.J. Budd & M.D. Piggott, "Geometric integration and its applications", Technical Report,University of Bath (2002).
2. E. Hairer, Ch. Lubich & G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations", Springer Verlag, Berlin (2002).
A great deal of useful material on geometric integration can be found at the website of the Geometric Integration Interest Group of FoCM.
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