Abstract In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O ( h 4 + τ 2 ) in l ∞ -norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations.

中文翻译：

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非线性狄拉克方程两个保守四阶紧有限差分格式的最优逐点误差估计

摘要 本文针对具有周期性边界条件的(1+1)维非线性狄拉克方程，提出并分析了两种保守的四阶紧致有限差分格式。基于矩阵知识，我们将所提出的紧凑方案的逐点形式转换为等效的向量形式，并分析它们的保守性和收敛性。我们证明了所提出的方案在离散水平上保持了总质量和能量，并且方案的收敛速度在没有任何网格比限制的情况下，在 l ∞ -范数中处于 O ( h 4 + τ 2 ) 的数量级，其中 h 和 τ 分别是空间和时间步长。误差分析技术包括能量法和非线性截断技术或数学归纳法来限制数值近似解。进行数值实验以证实我们的理论分析。本文的研究方法可以很容易地扩展到高阶紧致方案或其他类型的波动方程。