Analysis of a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation

Abstract

In this paper, we propose and analyze a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise form of the proposed compact scheme into equivalent vector form and analyze its conservative and convergence properties. We prove that the new scheme conserves the total mass and energy in the discrete level and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of $$O(h^4 +\tau ^2)$$ in $$l^\infty $$ -norm, where h and $$\tau $$ are spatial and temporal steps, respectively. The techniques for error analysis include the energy method and the mathematical induction. The numerical experiments are carried out to confirm our theoretical analysis.