A higher-order compact (HOC) discretization of generalized 3D convection-diffusion equation (CDE) in nonuniform grid is presented. Even in the presence of cross-derivative terms, the discretization uses only nineteen point stencil. Extension of this newly proposed discretization to semi-linear and convection-diffusion-reaction problems is seen to be straightforward and this inherent advantage is thoroughly exploited. The scheme being designed on a transformation free coordinate system is found to be efficient in capturing boundary layers and preserve the nonoscillatory property of the solution. The proposed method is tested using several benchmark linear and nonlinear problems from the literature. Additionally, problems with sharp gradients are solved. These diverse numerical examples demonstrate the accuracy and efficiency of the scheme proposed. Further, the numerical rate of convergence is seen to approach four confirming theoretical estimation.

提出了非均匀网格中广义 3D 对流扩散方程 (CDE) 的高阶紧致 (HOC) 离散化。即使存在交叉导数项，离散化也仅使用 19 点模板。将这种新提出的离散化扩展到半线性和对流扩散反应问题看起来很简单，并且这种固有优势得到了充分利用。发现在自由变换坐标系上设计的方案在捕获边界层方面是有效的，并保留了解的非振荡特性。使用文献中的几个基准线性和非线性问题对所提出的方法进行了测试。此外，解决了陡峭梯度的问题。这些不同的数值例子证明了所提出方案的准确性和效率。