In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge–Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm’s stability, one is based on the Godunov–Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results.

在本文中，我们研究了抛物线方程的高阶紧致有限差分方法的数值边界处理的稳定性。紧凑的有限差分方案可以用相对较小的模板实现非常高的阶精度。为了匹配内部域中紧凑方案的收敛顺序，我们采用简化的逆 Lax-Wendroff (SILW) 程序（Tan 等人，2012 年；Li 等人，2017 年）作为我们的数值边界处理。三阶总变差递减 (TVD) Runge-Kutta 方法（Shu 和 Osher，1988 年）被用作我们在完全离散情况下的时间步长方法。采用两种分析技术来检查算法的稳定性，一种是基于 Godunov-Ryabenkii 理论，另一种是特征值谱可视化方法（Vilar 和 Shu，2015）。研究了半离散和完全离散的情况，这两种不同的分析技术产生了一致的结果。显示了几个数值实验结果来验证理论结果。