Design And Implementation Of Energy-preserving Algorithms For KGS Equation

The coupled Klein-Gordon-Schrodinger (KGS) equation is an important kind of partial differential equations (PDEs), which plays significant roles in the quantum field theory. Energy conservation is one of its intrinsic features. The thesis is devoted to develop the energy-preserving algorithm for the KGS equation. In recent years, the energy-preserving algorithm for ordinary differential equations (ODEs) is very hot,and there emerge many novel methods such as the average vector field method and Hamiltonian boundary value methods. In the current work, we will use these newly obtained energy-preserving algorithms for ODEs to construct numerical algorithms for PDEs. Firstly, we present the infinite dimensional Hamiltonian system of the KGS equation, and obtain the associated properties. Secondly, we apply the discrete singu-lar convolution method in space, and derive the semi-discrete ODE system as well as the corresponding Hamiltonian system. For the temporal discretization, we adopt the average vector field method, finite difference method and the Hamiltonian boundary value method, respectively. Thereafter, we obtain several fully discrete schemes for the KGS equation and strictly prove the energy conservation laws of these schemes theoretically. Finally, numerical experiments are presented to verify the theoretical analysis and illustrate the effectiveness of the proposed energy-preserving methods.