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Geometric numerical integration and Schrödinger equations. (English) Zbl 1239.65078
Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-100-2/pbk). viii, 138 p. (2012).
The rough contents of this lecture are as follows: Ch. 1: Introduction, Ch. 2: Finite dimensional backward error analysis, Ch. 3: Infinite dimensional and semi-discrete Hamiltonian flow, Ch. 4: Convergence results, Ch. 5: Modified energy in the linear case, Ch. 6: Modified energy in the semi-linear case, Ch. 7: Introduction to long time analysis, Bibliography and an Index.
The author is concerned with the geometric numerical integration of linear and semi-linear Schrödinger equations with polynomial nonlinearity supplied with periodic boundary conditions. For the discretization of the space derivatives he uses a Fourier pseudospectral method and for the discretization of time derivative he considers some symplectic splitting schemes. The crucial issue of analysis is the degree to which these numerical schemes qualitatively reproduce properties of the Hamiltonian partial differential equation. Over long time periods the Hamiltonian preservation and energy exchange between the eigenmodes of the solutions are observed.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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