Local Discontinuous Galerkin Method For Klein-Gordon-Schr(?)dinger Equation And Its Algorithmic Design

The Local discontinuous Galerkin method(LDG)method is applied to the Klein-Gordon-Schr(?)dinger(KGS)equation,and the semi-discrete scheme and the full-discrete scheme are constructed by using the linear numerical flux respectively.The energy conservation and mass conservation characteristics of these formats are studied and the corresponding executable formats are designed.The thesis is organized as follows:The first chapter introduces the background of the KGS equation,and then introduces the knowledge of discontinuous finite element theory and some related knowledge used in this dissertation.In the second chapter,two lemmas about boundary numerical fluxes are proposed,and it is proved that the semi-discrete scheme with only spatial direction dispersion under special boundary numerical flux conditions satisfies charge conservation and energy conservation.On this basis,the time direction is discretized to obtain two fully discrete schemes,and it is proved that the two formats still maintain the dual physical quantity conservation.In the third chapter,a set of general-purpose algorithms with matrix as the core operation is designed for the scheme of the KGS equation.The algorithm is easy to understand and efficient.Firstly,based on the characteristics of the full discrete scheme,the Legendre polynomial is used as the common basis function.Secondly,the three integral types and the flow module are analyzed separately.Finally,all the modules,under the uniform grid,are combined to obtain two formats that are convenient for computer execution.In the fourth chapter,the algorithm is applied to two constructed fully discrete conservation formats.The numerical experiments of two exact solutions are carried out respectively,and the spatial convergence order and energy conservation are investigated respectively.The experimental results are in line with expectations,indicate that the two fully discrete LDG scheme constructed in this thesis maintain conservation of energy and conservation of mass,and also show that the constructed algorithm has certain feasibility.Finally,the content of this dissertation is briefly summarized,and the future research ideas are proposed.