Splitting High-order Compact Difference Schemes For Deterministic/Stochastic Gross-Pitaevskii Equations

This thesis is devoting to designing some splitting high-order compact difference schemes for the deterministic and stochastic Gross-Pitaevskii equations with the effective of rotating angular momentum.It is very difficult to numerically solve the Gross-Pitaevskii equation,which is combined with multiple physical effects,especially multi-dimensional problems.For this reason,it is divided into linear subproblems and nonlinear sub-problems by the splitting methods.Then we use the local one-dimensional method to divide the multi-dimensional linear sub-problems into formal one-dimensional problems.The nonlinear sub-problems can be exactly solved because of the pointwise conservation law of mass,and linear sub-problems apply high-order compact methods for discretization.It proves the stability and discrete mass conservation law.Finally,numerical experiments verify the efficiency,mass conservation and error accuracy of the schemes.The outline of this thesis is planned as follows:Chapter 1 mainly investigates and analyzes the physical background and current research status at home and abroad of the deterministic and stochastic GrossPitaevskii equation with angular momentum rotation.Chapter 2 studies some preliminary knowledge that needs to be used in the process of constructing numerical schemes,mainly including high-order compact method,time splitting method,and local one-dimensional method.Chapter 3 constructs the splitting high-order compact difference schemes for the deterministic Gross-Pitaevskii equation.The basic idea is to use the splitting method to divide it into first-order and second-order convergence sub-problems.The time direction applies the Crank-Nicolson method,and the space direction is discretized by high-order compact method.Then we can construct the firstorder splitting high-order compact method and the second-order splitting high-order compact method.Studies prove these numerical schemes are stable and conserve the total mass.Finally,the numerical examples are used to verify the efficiency of the schemes and the influence of different parameters on the accuracy and error order,simulate the waveform diagram of the Gross-Pitaevskii equation at different times,and verify the theoretical results of the schemes.Chapter 4 firstly conducts a theoretical analysis of the stochastic term for the stochastic Gross-Pitaevskii equation and proves that it can be solved accurately and satisfy the conservation of mass.Then we construct a splitting high-order compact difference scheme.The basic idea is to use high-order compact method to discretize the spatial direction,and use the Crank-Nicolson method to discretize the time direction.Studies have shown that these numerical schemes are stable and conserve the total mass.Finally,numerical experiments are used to verify the error convergence order and theoretical results of the schemes,and the influence of different noises on the waveform of the stochastic Gross-Pitaevskii equation is simulated.In Chapter 5,some conclusions are summarized based on the theoretical analysis and numerical illustration.The prospect work to be done is planned.