We propose an optimal mass transport method for the random genetic drift problem driven by a Moran process formulated as a degenerate reaction-advection-diffusion equation. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Dirac-delta singularities for genetic segregation on one hand, and preserves several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties, and to demonstrate the spatio-temporal dynamics of random generic drift. This talk is based on a joint work with Jose A. Carrillo and Lin Chen.