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.

在本文中，我们提出并分析了具有周期性边界条件的Klein-Gordon-Dirac方程的保守四阶紧致有限差分格式。基于矩阵知识，我们将拟议的紧致方案的逐点形式转换为等效向量形式，并分析其保守性和收敛性。我们证明了该新方案在离散级上节省了总质量和能量，并且该方案的收敛速度不受网格比率的限制，约为\（O（h ^ 4 + \ tau ^ 2） \）在（（l ^ \ infty \）- norm中，其中h和\（\ tau \）分别是空间和时间上的步骤。误差分析技术包括能量法和数学归纳法。进行数值实验以证实我们的理论分析。