By Erwan Faou
The objective of geometric numerical integration is the simulation of evolution equations owning geometric houses over lengthy occasions. Of specific value are Hamiltonian partial differential equations in general coming up in software fields resembling quantum mechanics or wave propagation phenomena. They show many vital dynamical positive factors akin to power upkeep and conservation of adiabatic invariants over very long time. during this atmosphere, a usual query is how and to which volume the replica of such very long time qualitative habit might be ensured via numerical schemes.
Starting from numerical examples, those notes supply an in depth research of the Schrödinger equation in an easy surroundings (periodic boundary stipulations, polynomial nonlinearities) approximated via symplectic splitting equipment. research of balance and instability phenomena prompted by means of area and time discretization are given, and rigorous mathematical reasons for them.
The ebook grew out of a graduate point path and is of curiosity to researchers and scholars looking an advent to the subject material.
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Extra resources for Geometric Numerical Integration and Schrodinger Equations
28). 7. Let P 2 Pk for some given k 2 N, M > 0 and s 0. Then there t s exists t and for all jtj Ä t a mapping 'H W BM ! z 0 / is real for jtj Ä t . 28). If moreover z 0 D . 0 ; N 0 / is real, then 'H s and tN > 0, and let us consider the Banach space E D Proof. Œ tN; tN; `s / equipped with the norm k . /k E D sup 2 Œ tN;tN k . z 0 / C t 0 'Tt ı XP . // d ; defines an mapping T W E 7! E. 6. Now consider the function Œ tN; tN 3 t 7! t/ 2 BM 0 2 E. Let Á > 0, and consider the ball in E, centered in 0 and with radius Á: o n 0 Ä Á : .
The existence of Z then follows from standard ODE arguments. 0/ D 0, holds for small t > 0. 3 Recursive equations. 13) and the decomposition H D T C P . 7). p; q/ and the time t. 20) can be seen as a nonlinear transport equation. 0; /. Indeed, we cannot find a norm on nonlinear Hamiltonian functions satisfying kfH; Kgk Ä C kH k kKk for a uniform constant C independent on H and K. This is due to the presence of derivatives in the Poisson brackets. 20) is in general divergent. Note however that this is possible in the class of quadratic Hamiltonians of the form H D y T Ay for some symmetric matrix A.
Note moreover that all the Z` are made of multiple derivatives of the Hamiltonian functions T and P . 20) up to a small error of Z . T C P / C N `D2 N order O. / with a constant depending on N . 13. Assume that T and P are smooth Hamiltonian functions and H D T C P . Let N 2 N and M > 0 and 0 be fixed. y/ Ä C N C1 : 37 3 Backward analysis for splitting methods Proof. For all > 0 and N 2 N, we define the function Z . 22), and we set H N D 1 Z N . /. 24). y/ in the sense of formal series in powers of t.
Geometric Numerical Integration and Schrodinger Equations by Erwan Faou