An SDG Galerkin structure‐preserving scheme for the Klein‐Gordon‐Schrödinger equation

Abstract

In this paper, we use the Galerkin weak form to construct a structure‐preserving scheme for Klein‐Gordon‐Schrödinger equation and analyze its conservative and convergent properties. We first discretize the underlying equation in space direction via a selected finite element method, and the Hamiltonian partial differential equation can be casted into Hamiltonian ordinary differential equations based on the weak form of the system afterwards. Then, the resulted ordinary differential equations are solved by the symmetric discrete gradient method, which yields a charge‐preserving and energy‐preserving scheme. Moreover, the numerical solution of the proposed scheme is proved to be bounded in the discrete norm and convergent with the convergence order of in the discrete norm without any grid ratio restrictions, where and are space and time step, respectively. Numerical experiments conducted last to verify the theoretical analysis.