This paper establishes a unified framework for the space-time convergence analysis of the energy-stable third-order accurate exponential time differencing Runge-Kutta schemes. By employing Fourier pseudo-spectral discretization in space and the inner product technique, we derive a rigorous Fourier eigenvalue analysis, which provides a detailed optimal convergence rate and error estimate. The primary challenge is addressing the complex nonlinear terms in the NSS equation. Fortunately, this challenge could be resolved through careful eigenvalue bound estimates for various operators.

翻译：本文为能量稳定的三阶指数时间差分Runge-Kutta格式建立了一个时空收敛性分析的统一框架。通过采用空间上的傅里叶伪谱离散化和内积技术，我们推导出严格的傅里叶特征值分析，从而提供了详细的最优收敛速率和误差估计。主要挑战在于处理NSS方程中的复杂非线性项。幸运的是，通过对各类算子进行精细的特征值界估计，这一难题得以解决。