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A finite proximate method was presented to solve the convection-diffusion equation in curvilinear grids. The method has characteristics of automatic upwind effect and the good stability. It was verified through exact solution and other calculation results of two-dimensional dam-break flow in a frictionless, horizontal channel. The calculation results are in good agreement with the exact solution and other calculation results, which show that the finite proximate method can be applied to solve the convection-diffusion equation directly not only in the rectangular grids, but also in the curvilinear grids.