Efficient Structure-Preserving Algorithms For Coupled Gross-Pitaevskii Equations

Gross-Pitaevskii equation(GPE)is an important class of equations in the quantum physics and is generally used to describe the properties of a single Bose-Einstein condensation(BEC).To describe the case of two or more BECs,the coupled Gross-Pitaevskii equations(CGPEs)are general used.The equations contains nonlinear terms and has both first-order and second-order spatial derivatives,the parameter of first-order spatial derivatives term are dependent on spatial variables,which makes it difficult to abtain the solution of CGPEs efficiently.Based on the energy conservation of CGPEs,the thesis try to use discrete cosine pseudo-spectral method,radial basis function(RBF)method and(combined)high-order compact method to construct the structure-preserving algorithms.All these methods have high convergence accuracy and lower computational cost.Finally,the convergence of these schemes and energy conservation law are verified by numerical experiments.The outline of the thesis is as follows:In Chapter 1,first analyzes the research status and physical background of GPE and CGPEs at home and abroad,then discusses the necessity of the research,and provides preliminary knowledge needed in the process of constructing the structure-preserving schemes,mainly including energy-preserving algorithm.In Chapter 2,the discrete cosine pseudo-spectral method is introuduced to construct a semi-discrete scheme with high-order spatial accuracy,and using Crank-Nickson method in time direction which is a two-level energy-preserving scheme.Simultaneously,when nonlinear terms in CGPEs are considered,the computation cost of iterating the nonlinear equations generated when solving the implicit scheme are quite expensive.In order to solve the problem,we construct a three-level explict scheme that reduces the computational cost significantly,and the projection method is introduced to maintain the energy conservation of the scheme.In Chapter 3,the radial basis function collocation(RBFC)method is used to construct the structure preserving algorithm of CGPEs.RBFC method is a meshless method,which has great advantages in dealing with non-regular boundaries and deform boundaries,and is less sensitive to the selection of free parameters in radial basis function.Based on the adventagement of RBFC methods,a energy-preserving scheme is constructed by using local RBFC(LRBFC)method and global RBFC(GRBFC)method for two dimensions inrotation CGPEs,and a symplectic-preserving scheme of CGPEs is proposed.In Chapter 4,we uses high-order compact method and combined high-order compact method to construct three-point-fourth-order and three-point-sixth-order numerical scheme for CGPEs.The combined high-order compacting method maintains the efficiency of the high-order compact method for problems with multiple derivatives of first and second order or more,combined high-order compact scheme maintains the high efficiency of the high-order compact method and improve the spatial accuracy to sixth order.Numerical experiments have verified the high accuracy of the scheme.Finally,some conclusions are summarized based on the theoretical analysis and numerical experiments.and the prospect work to be done is planned.