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SPH has a reasonable mathematical background. Although VBM and MPS are similar to SPH, their ma-thematical backgrounds seem fragile. VBM has some problems in treating the viscous diffusion of vortices but is known as a practical method for calculating viscous flows. The mathematical background of MPS is also not sufficient. Not with standing, the numerical results seem reasonable in many cases. The problem common in both VBM and MPS is that the space derivatives necessary for calculating viscous diffusion are not estimated reasonably, although the treatment of advection is mathematically correct. This paper discusses a method to estimate the above mentioned problem of how to treat the space derivatives. The numerical results show the comparison among FDM (Finite Difference Method), SPH and MPS in detail. In some cases, there are big differences among them. An extension of SPH is also given.

Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS)