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中文摘要：
重点讨论了LSFD（least square-based finite difference）方法和传统的FD（finite difference）方法在性能上的对比问题．对于传统的中心差分格式，一阶导数和二阶导数在二维情况的数值格式基架点有9个点，三维情况有27个点．在同样的基架点下，给出了LSFD方法近似一阶导数和二阶导数的显式公式，并指出LSFD方法在这种情况下实质上就是在不同网格线上的传统中心差分格式的组合．在数值模拟中，LSFD方法达到收敛所需要的迭代步数比传统差分格式少，并且x和y方向的网格纵横尺度比在LSFD方法中是一个非常重要的参数，对计算的稳定性有重要影响．
英文摘要：
The performance of the LSFD（least square-based finite difference） method is compared with the conventional FD（finite difference） schemes. For the approximation of the first and second order derivatives by the conventional central difference schemes, 9-point stencils for the 2D case and 27-point stencils for the 3D case are usually used. When the same stencils are used, the explicit LSFD formulations for approximation of the first and second order derivatives were present. The LSFD formulations are actually the combination of conventional central difference schemes along relevant mesh lines. It is found that LSFD formulations need much less iteration steps than the conventional FD schemes to get the converged solution, and the ratio of mesh spacing in the x and y directions is an important parameter in the LSFD application, which has a great effect on the stability of LSFD computation.
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Numerical simulation of two dimensional granular shear flows and the friction force of a moving slab
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