An AVF Galerkin Structure-Preserving Algorithm for the Klein-Gordon-Schrödinger Equation

Source abstract

Based on the notion that the numerical technique should consider and examine the conservation and convergence of the inherent properties for the original system, we construct and analyze a structure-preserving algorithm for the Klein-Gordon-Schrödinger equation using the averaged vector field Galerkin spectral element method in this paper. First, we discretize the fundamental equation in spatial direction via the spectral element method based on the weak form of the system, which yields a Hamiltonian ordinary differential equation. Second, the averaged vector field method is used to solve the resulted Hamiltonian system to obtain an energy-preserving and numerically charge-preserving scheme for the Klein-Gordon-Schrödinger equation. The higher-order fully-discrete schemes in time direction are also obtained by applying the corresponding higher-order averaged vector field method. Then, the error estimate is given with the convergence order of $\mathcal{O}(\Deltat^2 + \Deltax^{(min(N,r)}N^{−1})$ in the discrete $L^2$-norm, where $N$ is the order of the cardinal basis functions, $r$ represents the order of the Soblev space, $\Deltat$ and $\Deltax$ are time and space steps, respectively. Finally, numerical experiments are conducted to verify the theoretical analysis well.