High Order Compact Splitting Multisymplectic Schemes For The Nonlinear Schr(?)dinger Equations

In the thesis, we mainly develop some novel solvers including high order compact alternating direc-tion implicit (HOC-ADI) scheme, high order compact (HOC) splitting scheme, symplectic Fourier pseudo-spectral integrator and high order compact splitting multi-symplectic (HOC-SMS) scheme for di?erentkinds of Schro¨dinger equations in the hope of higher accuracy and more time-saving. We analyze theirproperties like stability, discrete conservation laws and symplectic structure-preserving, then detailed nu-merical results confirm with our theoretical conclusions.In Chapter 2, we firstly introduce some basic knowledge about symplectic geometry and symplecticspace. Then a family of common methods for the di?erent Schro¨dinger equations are shown, such as high-order compact (HOC) scheme, splitting approach, alternating direction implicit (ADI) scheme and sym-plectic method. Lastly, we study the preliminaries about symplectic integrator and Runge-Kutta method.In Chapter 3, an HOC-ADI scheme is firstly designed for the two-dimensional linear Schro¨dingerequation (LSE). It is proved that the scheme is unconditionally stable and preserves two discrete conserva-tion laws. Then, we extend the method to NLS equations. Furthermore, the HOC technique coupled withDouglas ADI method is generalized to three-dimension LSEs which is also unconditionally stable. Lastly,detailed numerical examples illustrate that the schemes are e?cient in time saving and high accuracy, whichare consistent with our theoretical analysis.In Chapter 4, symplectic Fourier pseudo-spectral integrators for the Klein-Gordon-Schro¨dinger equa-tions (KGS) are investigated. A di?erent Hamiltonian formulation from other literatures is presented.The Fourier pseudo-spectral discretization is applied to the space approximation which leads to a finite-dimensional Hamiltonian system. Then the symplectic integrators, including Sto¨rmer/Verlet method andmidpoint rule are adopted to time direction which lead to symplectic integrator for the KGS, respectively. Itis suggested that the Sto¨rmer/Verlet method is explicit which can be coed e?ciently, and the midpoint rulecaptures the mass of the original system exactly. Numerical experiments show that the symplectic integratorcan simulate various solitary very well over a long period.In Chapter 5, we propose a high-order compact splitting multi-symplectic (HOC-SMS) scheme for thecoupled nonlinear Schro¨dinger (CNLS) equation. It is analyzed that this scheme is unconditionally stableand achieves sixth order accuracy in spatial direction. It satisfies a system of conservation laws including themulti-symplectic conservation law, charge, energy, momentum invariants. Splitting method contributes totime saving as well. The validity and e?ciency of the scheme are tested by various numerical experiments,moreover, detailed numerical results are consistent with the theoretical conclusions.