Two fourth‐order compact finite difference schemes including a Crank–Nicolson one and a semi‐implicit one are derived for solving the nonlinear Klein–Gordon equations in the nonrelativistic regime. The optimal error estimates and the strategy in choosing time step are rigorously analyzed, and the energy conservation in the discrete sense is also studied. Under proper assumption on the analytical solutions, the errors of the two schemes both are proved to be of with mesh size h and time‐step τ. Numerical simulations are provided to confirm the theoretical analysis.

中文翻译：

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非相对论状态下非线性Klein-Gordon方程四阶紧致有限差分方法的最优误差估计

推导了两种四阶紧致有限差分方案，包括Crank-Nicolson一阶和半隐式一阶微分有限差分方案，用于求解非相对论状态下的非线性Klein-Gordon方程。严格分析了最优误差估计和时间步长选择策略，并研究了离散意义上的节能问题。在解析解的适当假设下，两种方案的误差都被证明是具有网格大小h和时间步长τ的。提供数值模拟以确认理论分析。