  We propose and analyze an optimal mass transport method for a random genetic
drift problem driven by a Moran process under weak-selection. The continuum
limit, formulated as a reaction-advection-diffusion equation known as the
Kimura equation, inherits degenerate diffusion from the discrete stochastic
process that conveys to the blow-up into Dirac-delta singularities hence brings
great challenges to both the analytical and numerical studies. 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.
