Abstract: The application of the finite difference method follows the discretization criterion of the discrete equivalent equation in the body fitted coordinate system, and the coordinate transformation coefficient of the local grid points is used in the scheme, enabling the creation of a new finite difference method for unstructured grids. The numerical results show that the first-order upwind scheme constructed by the three-grid lines connecting discrete points can converge stably in two dimensional space. In this paper, we extend this unstructured grid finite difference method to three dimensions. The discrete calculation is performed on four lines, and a new algorithm based on multiple grids in the local area and repetitive calculation is then proposed. Firstly, the calculation is applied to the whole region, which contains the object, using the cartesian-coordinated uniform grid. The second step is to eliminate the influence on the external flow field from the internal points of the object. Finally, the unstructured grid is filled in the small local area between the object surface and the rectangle grid to conduct the calculation. This calculation process is similar to Chinese traditional tie-dye process. Compared with the structured grid finite difference method, the tie-dye algorithm contains some useless grid points which require special treatment, and some local areas are repeatedly calculated. However, its advantages include simple programming, small memory occupation, fast grid-generation, and easy grid-expansion, at the cost of computing capacity waste. Qualitative analysis shows that this tie-dye algorithm is suitable for the development of the parallel computing in a large scale. The computational fluid dynamics application based on this algorithm can fully utilize the efficiency of the supercomputers with hundred thousand to millions of processor cells.

Key words: finite difference methods, unstructured grids, discrete equivalence, large scale parallel computing, tie-dye algorithm