Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein–Gordon–Schrödinger equations

The Klein–Gordon–Schrödinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schrödinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schrödinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.