Due to the fact that the boundary value solution of the biharmonic equations is one of the hot spots in the research of the boundary value solution of elliptic equations, a fourth-order compact difference scheme has therefore been established, thus theoretically verifying the uniqueness, stability and convergence of the difference solution. On this basis, a sixth-order extrapolated compact difference scheme can be obtained by a one-time extrapolation with Richardson extrapolation method adopted. Finally, numerical examples of two biharmonic equations are used to verify the effectiveness of this difference scheme on biharmonic equations.