A new numerical method named as basic function method is proposed. It can directly discretize differential operators on unstructured grids. By expanding the basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as a basic function and applying the flux splitting method and the combination of central and upwind schemes to suppress non-physical fluctuation near shock waves, a second-order basic function scheme of polynomial type is proposed to solve inviscid compressible flows numerically. Numerical results of typical examples for two-dimensional inviscid compressible transonic and supersonic steady flows indicate that the new scheme has high accuracy and high resolution for shock waves. Combined with the adaptive remeshing technique, satisfactory results can be obtained.