A new family of compact schemes of increased accuracy using quasi-variable mesh is presented for determining approximate solutions to the three-space dimensions mildly nonlinear convection dominated diffusion equations. The main thought behind the proposed scheme is to get uniformly distributed local truncation error, which otherwise not possible in case of finite-difference discretization using constant step-sizes mesh points. According to the zero or nonzero values of mesh stretching quantities, the increased accuracy fourth-order method refers to uniform meshes or quasi-variable meshes schemes. It is easy to tune the mesh points depending upon the location of subdomains having comparatively more fluctuating solution behavior. The block-tridiagonal construction of the Jacobian (iteration matrix) acquired with the new difference scheme makes it easier to implement with a reasonable computing time. We will describe the matrix and graph theoretic approach to analyze the convergence property and error estimate of the generalized scheme. A measure of accuracy, such as maximum errors, root-mean-squared errors, and numerical convergence rate, is examined by solving various forms of convection-dominated diffusion equations.

中文翻译：

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基于拟变网格的三维弱非线性静止对流-扩散方程的四阶紧格式

提出了一种使用准变量网格来提高精度的紧凑方案的新系列，用于确定三空间维温和非线性对流主导扩散方程的近似解。所提出方案背后的主要思想是获得均匀分布的局部截断误差，否则在使用恒定步长网格点的有限差分离散化的情况下这是不可能的。根据网格拉伸量的零值或非零值，提高精度的四阶方法是指均匀网格或准变网格方案。根据具有相对波动较大的求解行为的子域的位置来调整网格点很容易。用新的差分方案获得的雅可比（迭代矩阵）的块三对角构造使得在合理的计算时间下更容易实现。我们将描述矩阵和图论方法来分析广义方案的收敛性和误差估计。通过求解各种形式的以对流为主的扩散方程来检查精度的度量，例如最大误差、均方根误差和数值收敛速度。