High Order Compact Splitting Multisymplectic Schemes For Gross-pitaevskii Equation

In the thesis, we mainly develop some novel solvers including high order compa ct splitting multisymplectic scheme and high order compact local one-dimensional sc-heme for different dimensions of Gross-Pitaevskii(GP) equation in the hope of higher accuracy. We analyze their properties like stability, conservation laws and symplectic structure-preserving, then detailed results confirm with our theoretical conclusions.As we known, the form of GP equation the same as the nonlinear Schrodinger equation, and the nonlinear Schrodinger equation is the important model in physics. In the past several years, the researches of numerical methods of Schrodinger equation are various. On the basis, we develop a new numerical method, named high order co-mpact splitting multisymplectic scheme, and we through vast numerical experiments verify the new scheme is effective. The main contents of this paper are as follows:In Chapter1, we mainly introduce the related work of the GP equation in detail, such as the subject’s significance, introduction, and the paper’s main contents.In Chapter2, we mainly review the necessary theoretical knowledge, including some basic knowledge about symplectic geometry, high order compact scheme, splitt-ing approach, and multisymplectic Hamiltonian system.In Chapter3, we first design a new high order compact splitting multisymplectic scheme for one-dimensional GP equation. Then, it is analyzed that this scheme has unconditionally stability, achieves sixth order accuracy in spatial direction and a syst-em of conservation laws. Lastly, detailed numerical examples illustrate that the sche-me is efficient in time saving, high accuracy and long-time stable, which is consistent with our theoretical analysis.In Chapter4, we extend the GP equation to two-dimensional GP equation. Firstly, we propose a high order compact local one-dimensional splitting multisymplectic sch-eme for the two-dimensional GP equation. Then, analyze the theoretical properties of the new scheme. It is suggested that the scheme is unconditional stable, preserves discrete energy and mass conservation laws and LOD energy conservation law. Lastly, numerical experiments test that the numerical results confirm with our theoretical conclusions.In Chapter5, On the basis of previous chapter, the high order compact local one-dimensional scheme is generalized to three-dimension GP equation which is also unconditionally stable. Firstly, we develop the new scheme of the three-dimension GP equation. Secondly, we analyze the theoretical properties of numerical method in det-ail. Lastly, the validity and efficiency of the scheme are tested by various numerical experiments. Moreover, detailed numerical results are consistent with the theoretical conclusions.