معادله ي شرودينگر و روش هاي جدا سازي سيمپلكتيك

چكيده انگليسي :

Schr dinger equation and symplectic splitting methods IMAN DALILI SHOAEI i dalili@math iut ac ir June 19 2018 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mehdi Tatari mtatatri@cc iut ac irAdvisor Dr Majid Gazor mgazor@cc iut ac ir2010 MSC 65L20 65M06Keywords Splitting methods Hamiltonian system Symplectic map Spectral methods and Hermite interpolation Abstract This M Sc thesis is based on the following papers Sergio Blanes Fernando Casas Ander Murua On the linear stability of splitting methods Foundations of Computational Mathematics 8 2008 357 393 Sergio Blanes Fernando Casas Ander Murua Error analysis of splitting methods for the time dependent Schr dinger equation SIAM Journal on Scientific Computing 33 2011 5943 5949 Sergio Blanes Fernando Casas Ander Murua An efficient algorithm based on splitting for the time inte gration of the Schr dinger equation Journal of Computational Physics 303 2015 396 412Differential equations play an important role in applied mathematics and they exist in almost every science and en gineering field In many cases the differential equation modelling the physical phenomenon one aimes to studypossesses qualitative geometric properties like symplecticity volume preserving and etc that are absolutely es sential to preserve under discretization Hamiltonian systems constitude a clear example since their solution operatorsare symplectic The numerical integration of Hamiltonian systems by a conventional method leads to numerical flowsthat are not symplectic If the time interval is short and the integration scheme provides a reasonable accuracy theresult may be tolerable in practice However in many applications one needs to consider large time intervals sothat the computed solution is useless due to its lack of symplecticity Thus a necessity rises up to construct special purpose integrators that when applied to a Hamiltonian system do preserve the symplectic structure in the numericalflow These are known as symplectic integration algorithms and they not only outperform standard methods from aqualitative point of view but also the numerical error accumulates more slowly which is very important in long timecomputations The search for numerical integration methods that preserve the geometric structure of the problem has been general ized to other types of differential equations possessing a special structure worth being preserved under discretization The design and analysis of numerical integrators preserving structure constitude the zone of geomtric numerical in tegration Briefly in geometric integration one is not only concerned with the classical accuracy and stability of thenumerical algorithm but also the method must include the geometric properties of the main problem This givesthe integrator an improved qualitative behavior and provides a significantly more accurate long time integration incomparison with general purpose methods In this thesis the aim is to use symplectic integrators to solve spatially discretized Schr dinger equation The wholeprocess in a few words is as follows In Chapter 1 two popular techniques namely composition and splitting have been reviewed owing to the fact thatstructure preserving methods of desired order can be constructed by means of them Then Hamiltonian systems andtheir symplectic flows are discussed

دليلي شعاعي، ايمان

معادله ي شرودينگر و روش هاي جدا سازي سيمپلكتيك

روش هاي جداسازي , سيستم هاي هميلتوني , نگاشت سيمپلكتيك , روش هاي طيفي , درون يابي هرميتي