Optimal error estimates of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

Abstract

Recently, a time-splitting Fourier pseudo-spectral (TSFP) scheme for solving numerically the Klein–Gordon–Dirac equation (KGDE) has been proposed (Yi et al., 2019). However, that paper only gives numerical experiments and lacks rigorous convergence analysis and error estimates for the scheme. In addition, the time symmetry of the scheme has not been proved. This is not satisfactory from the perspective of geometric numerical integration. In this paper, we proposed a new TSFP scheme for the KGDE with periodic boundary conditions by reformulating the Klein–Gordon part into a relativistic nonlinear Schrödinger equation. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. We make a rigorously convergence analysis and establish error estimates by comparing semi-discretization and full-discretization using the mathematical induction. The convergence rate of the scheme is proved to be second-order in time and spectral-order in space, respectively, in a generic norm under the specific regularity conditions. The numerical experiments support our theoretical analysis. The conclusion is also applicable to high-dimensional problems under sufficient regular conditions. Our scheme can also serve as a reference for solving some other coupled equations or systems such as Klein–Gordon–Schrödinger equation.