In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.

中文翻译：

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任意尺寸的某些PDE的高阶紧凑有限差分

在本文中，我们首先给出了具有可变对流系数的对流扩散方程的四阶紧凑有限差分（CFD）方案的模型表达式。然后，我们还获得了具有可变扩散系数的扩散方程的四阶CFD格式。此外，还对具有恒定系数的方程式开发了六阶CFD方案的详细描述，该方程式用于讨论具有任意维数的某些偏微分方程（PDE）。本文准备了各种数值测试计算方法来分别评估四阶CFD方案和六阶CFD方案的性能，并通过实验结果证明了该方案的有效性。