For the Klein-Gordon-Dirac equation (KGDE) with small coupling constant , we propose a Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method and establish the improved uniform error bounds in the time domain at . We first convert the KGDE to a coupled system and then consider LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric which is an important structure in numerical geometric integration. Through careful and rigorous convergence analysis, we establish the error bounds for the full-discretization, where is determined by the regularity conditions. If the solution is sufficiently smooth with sufficiently large, we obtain the errors with improved uniform bounds at in the long-time domain up to . These error bounds are much better than the classical bounds provided by the traditional analysis for the non-Lawson-type exponential wave integrators equipped with Fourier pseudo-spectral method. Combined with the classical analysis tools such as mathematical induction and energy method, we complete our error analysis by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation. By applying the LEWIFP method to some problems, we show the numerical results to support our error bounds. In addition, the numerical results also show that the discrete mass and energy are stable in the time domain which is long enough. Finally we extend our method to the oscillatory problem.

中文翻译：

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Klein-Gordon-Dirac 方程的 Lawson 型指数波积分器方法的改进均匀误差界


对于具有小耦合常数 的 Klein-Gordon-Dirac 方程（KGDE），我们提出了一种劳森型指数波积分傅立叶伪谱（LEWIFP）方法，并在 时域建立了改进的均匀误差界。我们首先将 KGDE 转换为耦合系统，然后考虑耦合系统的 LEWIFP 方法。 LEWIFP方法被证明是时间对称的，是数值几何积分中的一个重要结构。通过仔细而严格的收敛分析，我们建立了完全离散化的误差界限，其中由正则性条件决定。如果解足够平滑并且足够大，我们将在长期域中获得具有改进的统一边界的误差，直到 。这些误差界比传统分析为配备傅里叶伪谱方法的非劳森型指数波积分器提供的经典界好得多。结合数学归纳法、能量法等经典分析工具，采用规律性补偿振荡（RCO）技术完成误差分析，该技术通过解的规律性控制高频模式，通过相位抵消控制低频模式。通过将 LEWIFP 方法应用于一些问题，我们显示了数值结果来支持我们的误差范围。此外，数值结果还表明，离散质量和能量在足够长的时域内是稳定的。最后，我们将我们的方法扩展到振荡问题。