各种网格上统一的数值离散方法

提出一种在任意网格上计算数值微分的方法，这种方法利用各种不同网格所具有的共同性质， 基于Taylor展开和加权最小二乘法，得到了各种网格下都可以使用的数值微分格式. 有了这一技术， 可以极大地丰富已经发展起来的各种数值方法，原来只能用在结构网格上的格式，可以直接推广到 其他各种网格上，从而可以用于各种复杂区域内微分方程的数值求解. 初步的应用表明这种技术是 简单而有效的.

A universal numerical discretization method on different meshes

A universal discretization method is prestented in this paper, it can be used on arbitary meshes. Considering the common properties of all different kinds of meshes, we established this numerical difference scheme by Taylor series expansion and the least square method. We can obtain the local difference matrix(LDM) and global difference matrix(GDM) on any mesh by this method, then the difference operator can be interpreted as its matrix form in discreted space directly. This skill can apply to many numerical schemes developed on structured grid, then those schemes will work on arbitary meshes, the complexity of the computation domain will no longer be any problems. In order to verify the skill in this paper, we first compute the numerical difference of an analytical function, and compare the results with the exact solutions. It shows that the method has the 2nd order accuracy as the center difference scheme. Another two examples are numerical simulation of incompressible flow in a two dimension backward-facing step and three dimension driven cavity. The vorticity-stream function equation and Navier-Stokes equations in velocity-pressure form are used respectively. The results in this paper are agreement with the classical ones on structured meshes. But the method here can be applied on any grid.