As for the two-dimensional reaction-diffusion equation, a compact difference alternating direction implicit (ADI) scheme has been established, thus proving the uniqueness, stability, and convergence of the compact ADI scheme solution of the equation. Furthermore, by adopting Richardson extrapolation method, a first-order extrapolation difference scheme and a second-order extrapolation difference scheme can be established, with the errors and convergence orders of the established ADI scheme and extrapolation scheme verified through numerical examples. Based on a comparison of the numerical results of two schemes, it can be seen that Richardson extrapolation method can effectively improve the accuracy of numerical solutions and reduce errors.