An implicit-explicit relaxation extrapolated Runge–Kutta and energy-preserving finite element method for Klein–Gordon-Schrödinger equations

An implicit-explicit (IMEX) relaxation extrapolated Runge–Kutta (RERK) and energy-preserving finite element method is designed for Klein–Gordon-Schrödinger (KGS) equations with periodic boundary conditions. First, the RERK method is employed to the discretization in time, which can keep the unconditionally stable and energy conservation after selecting the appropriate relaxation parameters γn. An IMEX time-marching discretization is constructed so that only a system of linear equations needs to be solved at any time step. Next, the finite element method is utilized to the discretization in space. By the temporal-spatial splitting technique, optimal error estimates in the L2-norm and H1-norm are obtained with the convergent order O(τs+hk+1) and O(τs+hk), without any time stepsize restrictions. At last, some numerical examples are presented to demonstrate the theoretical results.