The wind field is often represented by stream function and velocity potential in the meteorological community. For a limited area, the finite-difference approach is used to calculate the stream function and velocity potential from the wind field. But there are often large departures between the original and reconstructed wind fields caused by the unsuitable boundary conditions. The basic finite-difference method for Arakawa A- and D-grids is introduced and the comprehensive discussion about the problems in the previous methods is given. Then a new method is presented, whose unique features include (a) the linear extrapolation of the original wind field and the extension to the computation domain, (b) the consistent finite-difference scheme, (c) the accurate calculations of non-divergent wind, irrotational wind and solution-determining condition of velocity potential at the boundary, and (d) the suitable grid distribution of variables. The correctness of the new method is proved mathematically. The new method can be also used for all other Arakawa grids. The results of experiments using the real data show the maximum magnitude of departures between the original and reconstructed wind fields is only 10-12 m/s, which is produced by the round-off error of the computer. The longstanding problem on accurate solution of the stream function and velocity potential from the wind field in a limited area by using the finite-difference approach, is perfectly solved in the present study.