This study aims to construct a stable, high-order compact finite difference method for solving Sobolev-type equations with Dirichlet boundary conditions in one-space dimension. Approximation of higher-order mixed derivatives in some specific Sobolev-type equations requires a bigger stencil information. One can approximate such derivatives on compact stencils, which are higher-order accurate and take less stencil information but are implicit and sparse. Spatial derivatives in this work are approximated using the sixth-order compact finite difference method (Compact6), while temporal derivatives are handled with the explicit forward Euler difference scheme. We examine the accuracy and convergence behavior of the proposed scheme. Using the von Neumann stability analysis, we establish $L_2-$stability theory for the linear case. We derive conditions under which fully discrete schemes are stable. Also, the amplification factor $\mathcal{C}(\theta)$ is analyzed to ensure the decay property over time. Real parts of $\mathcal{C}(\theta)$ lying on the negative real axis confirm the exponential decay of the solution. A series of numerical experiments were performed to verify the effectiveness of the proposed scheme. These tests include advection-free flow, and applications to the equal width equation, such as single solitary wave propagation, interactions of two and three solitary waves, undular bore formation, and the Benjamin-Bona-Mahony-Burgers equation.

翻译：本研究旨在构建一种稳定、高阶的紧致有限差分方法，用于求解一维空间中具有Dirichlet边界条件的Sobolev型方程。某些特定Sobolev型方程中高阶混合导数的近似需要较大的模板信息。这类导数可在紧致模板上近似，其具有高阶精度且所需模板信息较少，但为隐式且稀疏。本文采用六阶紧致有限差分法（Compact6）近似空间导数，而时间导数则通过显式前向欧拉差分格式处理。我们检验了所提格式的精度与收敛行为。利用von Neumann稳定性分析，我们为线性情形建立了$L_2-$稳定性理论。推导了确保全离散格式稳定的条件。同时，分析了放大因子$\mathcal{C}(\theta)$以保证其随时间衰减的特性。$\mathcal{C}(\theta)$实部落于负实轴证实了解的指数衰减。通过一系列数值实验验证了所提格式的有效性，包括无平流流动、等宽方程的应用（如单孤立波传播、双孤立波及三孤立波相互作用、涌波形成）以及Benjamin-Bona-Mahony-Burgers方程。